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Considered one of the founding fathers of integral geometry, Luis Santal\'o has contributed to various areas of mathematics. His work has applications in number theory, in the theory of differential equations, in stochastic geometry, in functional analysis, and also in theoretical physics. Between the 1950's and the 1970's, he wrote a series of papers on general relativity and on the attempts at generalizing Einstein's theory to formulate a unified field theory. His main contribution in this subject was to provide a classification theorem for the plethora of tensors that were populating Einstein's generalized theory. This paper, which is the transcript of the conferences delivered by the authors in occasion of the celebration of the 100th anniversary of Luis Santal\'o's birth, revisits his work on theoretical physics., Comment: 10 pages. This paper is the transcript of the conferences delivered by the authors in occasion of the celebration of the 100th anniversary of Luis Santal\'o's birth. v2 minor error corrected; version accepted for publication in EPJH

Hypothesis Emerging energy-related technologies deal with multiscale hierarchical structures, intricate surface morphology, non-axisymmetric interfaces, and complex contact lines where wetting is difficult to quantify with classical methods. We hypothesise that a universal description of wetting on multiscale surfaces can be developed by using integral geometry coupled to thermodynamic laws. The proposed approach separates the different hierarchy levels of physical description from the thermodynamic description, allowing for a universal description of wetting on multiscale surfaces. Theory and Simulations The theoretical framework is presented followed by application to limiting cases of wetting on multiscale surfaces. Limiting cases include those considered in the Wenzel, Cassie-Baxter, and wicking state models. Wetting characterisation of multiscale surfaces is explored by conducting simulations of a fluid droplet on a structurally rough surface and a chemically heterogeneous surface. Findings The underlying origin of the classical wetting models is shown to be rooted within the proposed theoretical framework. Integral geometry provides a topological-based wetting metric that is not contingent on any type of wetting state. The wetting metric is demonstrated to account for multiscale features along the common line in a scale consistent way; providing a universal description of wetting for multiscale surfaces.

Volterra integral equations of the first kind arise naturally in various problems. Many problems in mathematical physics, engineering, and integral geometry are often reduced to first-kind Volterra integral equations. A survey of regularization methods for first kind Volterra equations is given by Lamm. This chapter uses the Legendre wavelet to solve Volterra integral equations of the first kind. The aim is to present an efficient numerical method to find a stable solution of Volterra integral equations of the first kind. The chapter commences with an introduction to Legendre wavelets and an approximation of function using Legendre wavelets. It also introduces a local regularization method and presents the numerical method to find an approximate solution to the given problem. The convergence and error analysis is considered and some numerical examples are presented to verify the accuracy of the numerical method.

The translative intersection formula of integral geometry yields an expression for the mean Euler characteristic of a stationary random closed set intersected with a fixed observation window. We formulate this result in the setting of sets with positive reach and using flag measures which yield curvature measures as marginals. As an application, we consider excursion sets of stationary random fields with $C^{1,1}$ realizations, in particular, stationary Gaussian fields, and obtain results which extend those known from the literature., A new section (Polytopal window) has been added

Hypothesis The development of high-resolution in situ imaging has allowed contact angles to be measured directly inside porous materials. We evaluate the use of concepts in integral geometry to determine contact angle. Specifically, we test the hypothesis that it is possible to determine an average contact angle from measurements of the Gaussian curvature of the fluid/fluid meniscus using the Gauss-Bonnet theorem. Theory and simulation We show that it is not possible to unambiguously determine an average contact angle from the Gauss-Bonnet theorem. We instead present an approximate relationship: 2 π n ( 1 - cos θ ) = 4 π - ∫ κ G 12 dS 12 , where n is the number of closed loops of the three-phase contact line where phases 1 and 2 contact the surface, θ is the average contact angle, while κ G 12 is the Gaussian curvature of the fluid meniscus which is integrated over its surface S 12 . We then use the results of pore-scale lattice Boltzmann simulations to assess the accuracy of this approach to determine a representative contact angle for two-phase flow in porous media. Findings We show that in simple cases with a flat solid surface, the approximate expression works well. When applied to simulations on pore space images, the equation provides a robust estimate of contact angle, accurate to within 3 ° , when averaged over many fluid clusters, although individual values can have significant errors because of the approximations used in the calculation.

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and geometry. We demonstrate the viability of Cauchy's infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson, Comment: 21 pages, Real Analysis Exchange

We consider weighted Radon transforms $$R_W$$ along hyperplanes in $$\mathbb {R}^3$$ with strictly positive weights W. We construct an example of such a transform with non-trivial kernel $$\mathrm {Ker}R_W$$ in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight W is rotation invariant. In particular, by this result we continue studies of Quinto (J Math Anal Appl 91(2): 510–522, 1983), Markoe and Quinto (SIAM J Math Anal 16(5), 1114–1119, 1985), Boman (J Anal Math 61(1), 395–401, 1993) and Goncharov and Novikov (An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. arXiv:1709.04194v2 , 2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in $$\mathbb {R}^d, \, d\ge 3$$ .

In the article [11] of L. Kunyansky a symmetric integral identity for Bessel functions of the first and second kind was proved in order to obtain an explicit inversion formula for the spherical mean transform where our data is given on the unit sphere in $\Bbb R^{n}$. The aim of this paper is to prove an analogous symmetric integral identity in case where our data for the spherical mean transform is given on an ellipse $E$ in $\Bbb R^{2}$. For this, we will use the recent results obtained by H.S. Cohl and H.Volkmer in [7] for the expansions into eigenfunctions of Bessel functions of the first and second kind in elliptical coordinates., 16

Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensionsn≤ 4.

We develop an integral geometry of stationary Euler equations defining some function w on the Grassmannian of affine lines in $$\mathbb {R}^3$$ depending on a putative compactly supported solution (v; p) of the system and deduce some linear differential equations for w. We conjecture that $$w=0$$ everywhere and prove that this conjecture implies that $$v=0.$$

The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $k\in \{1,\ldots ,d-1\}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.

We give a survey on the mathematics of computerized tomography. We start with a short introduction to integral geometry, concentrating on inversion formulas, stability, and ranges. We then go over to inversion algorithms. We give a detailed analysis of the filtered backprojection algorithm in the light of the sampling theorem. We also describe the convergence properties of iterative algorithms. We shortly mention Fourier based algorithms and the recent progresses made in their accurate implementation. We conclude with the basics of algorithms for cone beam scanning which is the standard scanning mode in present days clinical practice.

We transfer the results of Part I related to the modified support theorem and the kernel description of the hyperplane Radon transform to totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the spherical mean transform for spheres through the origin. The assumptions for functions are formulated in integral terms and close to minimal.

The following two inversion methods for Radon-like transforms are widely used in integral geometry and related harmonic analysis. The first method invokes mean value operators in accordance with the classical Funk-Radon-Helgason scheme. The second one employs integrals of the potential type and polynomials of the Beltrami-Laplace operator. Applicability of these methods to the horospherical transform in the hyperbolic space was an open problem. In the present paper we solve this problem for $L^p$ functions in the maximal range of the parameter $p$ and for compactly supported smooth functions, respectively. The main tools are harmonic analysis in the hyperbolic space and associated fractional integrals., 41 pages

We first investigate the translative containment measure for convex domain K0 to contain, or to be contained in, the homothetic copy of another convex domain K1, i.e., given two convex domains K0, K1 of areas A0, A1, respectively, in the Euclidean plane ℝ2, is there a translation T so that t(TK1) ⊂ K0 or t(TK1) ⊃ K0 for t > 0? Via the translative kinematic formulas of Poincare and Blaschke in integral geometry, we estimate the symmetric mixed isohomothetic deficit σ2(K0,K1) ≡ A012 − A0A1, where A01 is the mixed area of K0 and K1. We obtain a sufficient condition for K0 to contain, or to be contained in, t(TK1). We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality.

We present a methodology for a numerical analysis of three-dimensional tomographic images in this paper. The methodology is based on integral geometry, topology, and morphological analysis methods. It involves calculating cumulative and non-cumulative pore size distributions of Minkowski functionals and Betti numbers. We investigated 13 samples in dry and wet (saturated beyond the field capacity) conditions within different horizons of the Phaeozem albic. For samples of the arable horizon, an increase in the Euler characteristic was observed in the process of wetting. For samples from the A2, AB and B2 horizons, the Euler-Poincare characteristic decreased during wetting. It has been proven that both Betti numbers (number of isolated pores and number of &ldquo, tunnels&rdquo, ) decrease with swelling of the AB and B2 horizons at a depth of 20&ndash, 90 cm. For samples from the arable horizon, another dependence was observed: A Betti number of zero increased first but decreased during wetting. Based on the change in topological characteristics, two methods of changing the topology of the void space of the soil were demonstrated. The above-described quantitative changes of proposed parameters of pore space tomographic images prove the possibility and progressiveness of their usage for the pore space transformation estimate.

One hundred years ago, Blaschke and Poincare derived exact closed-form formulas for integrals over SE(n) of Euler characteristics of intersections of n-dimensional bodies for \(n=2,3\). Three decades thereafter Chern extended these formulas to the n-dimensional case. These results form a core tool in the fields of convex geometry, integral geometry, geometric probability, and stochastic geometry. These results, often referred to as “Principal Kinematic Formulae,” are extended here to the case of integrals of set-indicator functions, resulting in inequalities for the case of non-convex bodies. These results are relevant to assessing the frequency of occurrence of collisions that occur in sample-based robot motion planning.

A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the explicit form of the distribution of the determinant. We use these result to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix product, and we further give explicit expressions for the largest Lyapunov exponent. Generalisations to the case of complex or quaternion entries are also given. For standard Gaussian matrices $X$, the full Lyapunov spectrum for products of random matrices $I_N + {1 \over c} X$ is computed in terms of a generalised hypergeometric function in general, and in terms of a single single integral involving a modified Bessel function for the largest Lyapunov exponent., Comment: 23 pages

investigate the symmetric mixed isoperimetric deficit Δ2( K 0, K 1) of domains K 0 and K 1 in the Euclidean plane R 2 via the known kinematic formulas of Poincare and Blaschke in integral geometry. Then we obtain symmetric mixed isoperimetric inequality and some Bonnesen-style symmetric mixed inequalities. One of those inequalities strengthens Kotlyars inequality. Finally, we obtain some reverse Bonnesen-style symmetric mixed inequalities that generalize the known Bottemas result.

In this paper, we construct an holomorphic extension of certain functions on ℤ n . We deduce a discrete Gutzmer’s formula for the discrete Fourier transform (see Theorem 4.4). In addition, we prove the Hardy-Littlewood inequality for the classical Radon transform on ℝ n (see Theorem 4.7). In this work, we use the techniques of the diophantine integral geometry and the estimate method in certain weighted L 2-spaces.

In the recent articles Paternain et al. (J. Differ Geom, 98:147–181, 2014, Invent Math 193:229–247, 2013), a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under the geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.

The objective of reconstructive integral geometry is to recover a function from its integrals over a set of subvarieties. A parametrix is a method of reconstruction of a function from its integral data up to a smoothing operator. In the simplest case, a parametrix recovers a function with a jump singularity along a curve (surface) up to a continuous function, which can be quite informative in medical imaging. We provide an explicit construction for a wide class of acquisition geometries. The case of photo-acoustic geometry is of special interest.

We characterize the image of exponential type functions under the discrete Radon transformRon the latticeZnof the Euclidean spaceRn n≥2. We also establish the generalization of Volberg's uncertainty principle onZn, which is proved by means of this characterization. The techniques of which we make use essentially in this paper are those of the Diophantine integral geometry as well as the Fourier analysis.

In recent years on the space of translation invariant continuous valuations there have been discovered several canonical structures. Some of them turned out to be important for applications in integral geometry. In this chapter we review the relevant background and the main properties of the following new structures: product, convolution, Fourier type transform, and pull-back and push-forward of valuations under linear maps.

In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S.R. Moghadasi has shown how the decomposition of the $N$-fold product of Lebesgue measure on $\mathbb R^n$ implied by matrix polar decomposition can be used to derive the Blaschke-Petkantschin decomposition of measure formula from integral geometry. We use known formulas from random matrix theory to give a simplified derivation of the decomposition of Lebesgue product measure implied by matrix polar decomposition, applying too to the cases of complex and real quaternion entries, and we give corresponding generalisations of the Blaschke--Petkantschin formula. A number of applications to random matrix theory and integral geometry are given, including to the calculation of the moments of the volume content of the convex hull of $k \le N+1$ points in $\mathbb R^N$, $\mathbb C^N$ or $\mathbb H^N$ with a Gaussian or uniform distribution., Comment: 16 pages

Analytic reconstruction of a function defined in an affine space from data of its integrals along lines or rays is in focus of the paper. Basic tools are the Fourier transform of homogeneous distributions and a self-duality equation in integral geometry. Three dimensional case is of special interest.

In these pages we compute the expectation of several functions of multi-variate complex polynomials. The common thread of all our outcomes is the basic technique used in their proofs. The used techniques combine essentially the unitary invariance of Bombieri-Weyl's Hermitian product and some elementary Integral Geometry. Using different combinations of these techniques we compute the expectation of the logarithm of the absolute value of an affine polynomial and we compute the expected value of Akatsuka Zeta Mahler's measure. As main consequences of these results and techniques, we show a probabilistic answer to question (d) in Armentano and Shub (2012) [2], concerning the complexity of one point homotopy, and an Arithmetic Poisson Formula for the multi-variate resultant. These two last statements and bounds are related to the complexity of algorithms for polynomial equation solving.

Let η t be a Poisson point process of intensity t ≥ 1 on some state space Y and let f be a non-negative symmetric function on Y k for some k ≥ 1 . Applying f to all k -tuples of distinct points of η t generates a point process ξ t on the positive real half-axis. The scaling limit of ξ t as t tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m -th smallest point of ξ t is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Ito chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k -flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.

We study the nontrivial solvability of the homogeneous problem of integral geometry on the family of spheres formed by sections of the (n − 1)-dimensional unit sphere by hyperplanes perpendicular to the generators of a given cone. By using a relationship between this problem and the Dirichlet problem for an ultrahyperbolic equation in a ball and a criterion for the failure of uniqueness of the solution of the latter problem in terms of zeros of classical Jacobi polynomials, we obtain a criterion for the uniqueness of the solution of the former problem.

Kernel-based, smoothed estimates of spatial variables are useful in exploratory analyses because they yield a clear visual image of geographic variability in the underlying variable. In this paper I suggest an approach for assessing the significance of peaks in the surface that result from the application of the smoothing kernel. The approach may also be thought of as a method for assessing the maximum among a set of suitably defined local statistics. Local statistics for data on a regular grid of cells are first defined by using a Gaussian kernel. Results from integral geometry are then used to find the probability that the maximum local statistic (M) exceeds a given critical value (M). Approximations are provided that make implementation of the approach straightforward. Future work will address several other issues associated with local statistics that have been defined in this way, including edge effects, and the effects of global spatial autocorrelation on the choice of critical value.

General properties of spatial weights models, in particular Markovian properties, are systema- tically investigated. The role of stationary spatial distribution, interpretable as an importance-centrality or prominence index, is emphasized. Spatial interaction models, and among them the gravity model, are classified with respect to the time reversal and aggregation invariance properties obeyed by the asso- ciated spatial weights. Nine examples, involving connectivity, flows and distance decay analysis, integral geometry and Dirichlet-Voronoi tessellations illustrate the main concepts, with a particular geometri- cal emphasis, and show how traditional, heuristic ingredients aimed at defining spatial weights can be recovered form general models.

Within this work quite old concepts from integral geometry are applied to classical equilibrium thermodynamics of two-phase systems. In addition to the area as basic interfacial quantity the full geometric characterization of the interface is used, which includes the two remaining Minkowski functionals, the mean curvature integral and the Euler Poincaré characteristic. The basic energetic characteristic of the interface (i.e. the interfacial tension) is extended by two additional properties: edge force as (up to a factor 4/π) the work necessary to form a right-angled edge of unit length, and item energy as the work to form an additional item in the phase morphology. Both quantities are of increasing importance, when going to micro- and nano-scales. They are subsequently used for interfaces of arbitrary shape to derive a relationship extending the classical Young-Laplace equation. The supplementary contribution is proportional to the Gaussian curvature, with the edge force as proportionality constant. Furthermore, both edge force and item energy are shown to be applicable to the description of crystal nucleation in liquids (extending the classical Becker Döring theory). It turns out, that even above the thermodynamic melting temperature stable nuclei can be present in the liquid phase. They immediately are able to grow when quenched to a temperature below a characteristic temperature. This temperature of spontaneous homogeneous nucleation is simply connected to the edge force, whereas the number of stable clusters per unit volume is dominated by the item energy. Finally, the additional energetic interfacial properties are used in a similar way to characterize the stability of emulsions.

Combinatorial integral geometry possesses some results that can be interpreted as belonging to the field of Geometric Tomography. The main purpose of the present paper is to present a case of parallel X-ray approach to tomography of random convex polygons. However, the Introduction reviews briefly some earlier results by the author that refer to reconstruction of (non-random) convex domains by means of a point X-ray. The main tool in treating the parallel X-rays is disintegrated Pleijel identity, or rather, its averaged version, whose derivation is represented in complete detail. The paper singles out a class of random polygons called tomography models, that offer essential advantages for the analysis. The definition of a tomography model is given in terms of stochastic independence. Fortunately, random translation-invariant Poisson processes of lines in IR2 suggest a class of examples. We recall that each such line process is determined by its rose of directions ρ(ϕ). For rather general ρ(ϕ), the number weighted typical polygon in the polygonal partition of the plane generated by the corresponding Poisson line process happens to be a tomography model. For general tomography models, a differential equation is derived for the Laplace transform for parallel X-rays, that rises several interesting computational problems.

In this article, we first discuss inversion methods of the Abel-type integral transforms. It is well known that the support theorem of the Radon transform does not hold unless the function decreases rapidly at infinity. We prove that this support theorem holds for L 1 functions if they are radial. The inversion methods of the Abel-type integrals play an important role to prove the support theorem for the radial functions. We also prove inversion formulae of the exterior problems for the Radon transform of the radial functions.

Summary An unbiased estimator of the Euler-Poincare characteristic (Euler number) of an arbitrary object or a random structure, respectively, is given. The estimator is based on joint observations of pairs of parallel section profiles. Thus the present paper extends the use of Sterio's disector from counting particles to determining the Euler number for a wide class of probes. The correctness of the given formulae is proved with mathematical strictness. Furthermore, the feasibility of the method is illustrated by an example from materials sciences.

We introduce a new analytic family of intertwining operators which include the Radon transform over matrix planes and its inverse. These operators generalize integral transformations introduced by Semyanistyi (Dokl. Akad. Nauk SSSR 134:536–539, [1960]) in his research related to the hyperplane Radon transform in ℝ n . We obtain an extended version of Fuglede’s formula, connecting generalized Semyanistyi’s integrals, Radon transforms and Riesz potentials on the space of real rectangular matrices. This result combined with the matrix analog of the Hilbert transform leads to variety of new explicit inversion formulas for the Radon transform of functions of matrix argument.

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T * M.

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures., 26 pages, some misprints have been corrected and some unclear statements have been clarified. In particular, the proof of Theorem 2.1 has been modified slightly, the statement and proof of Theorem 3.1 has been changed and the local extension of mixed volumes has been made precise

We study the problem of coverage in planar heterogeneous sensor networks. Coverage is a performance metric that quantifies how well a field of interest is monitored by the sensor deployment. To derive analytical expressions of coverage for heterogeneous sensor networks, we formulate the coverage problem as a set intersection problem, a problem studied in integral geometry. Compared to previous analytical results, our formulation allows us to consider a network model where sensors are deployed according to an arbitrary stochastic distribution; sensing areas of sensors need not follow the unit disk model but can have any arbitrary shape; sensors need not have an identical sensing capability. Furthermore, our formulation does not assume deployment of sensors over an infinite plane and, hence, our derivations do not suffer from the border effect problem arising in a bounded field of interest. We compare our theoretical results with the spatial Poisson approximation that is widely used in modeling coverage. By computing the Kullback-Leibler and total variation distance between the probability density functions derived via our theoretical results, the Poisson approximation, and the simulation, we show that our formulas provide a more accurate representation of the coverage in sensor networks. Finally, we provide examples of calculating network parameters such as the network size and sensing range in order to achieve a desired degree of coverage.

A triplet of function s for the statistical characterization of planar point patterns is introduced. They are related to the integral-geometric quantities area, boundary length and Euler number of patterns of discs centred at the given points. These functions are able to give information on the distribution of a given point pattern which the traditional summary statistics of point process theory do not offer and so can lead to an improved statistical description. The paper describes the statistical estimation of the new characteristics. Some examples illustrate their application in the exploratory analysis of point patterns of tree positions in forests, in comparison to results obtained by means of second-order and distance characteristics.

Predicting the relationship between the morphology of porous media and their physical properties, e.g, the conductivity, elasticity and permeability, is a long-standing problem and important to a range of applications from geophysics to materials science. Here, a set of four morphological measures, so-called Minkowski functionals, is defined which allows one to quantitatively characterize the shape of spatial structures, to optimally reconstruct porous media, and to accurately predict material properties. The method is based on integral geometry and Kac's theorem which relates the spectrum of the Laplace operator to the four Minkowski functionals. Analytic expressions for mean values of Minkowski functionals in Boolean models allow the definition of an effective shape of a grain in a system made up of a distribution of arbitrarily shaped constituents. Reconstructing the microstructure using this effective grain shape leads to an excellent match to the percolation thresholds and to the mechanical and transport properties across all phase fractions. Additionally, the use of the effective shape in effective medium formulations leads to good explicit predictions of bulk moduli. The method is verified for several model systems and sedimentary rock samples, demonstrating that a single tomographic image is sufficient to estimate the morphology and physical properties such as permeabilities and elastic moduli for a range of porosities. Also the thermodynamic behaviour of fluids in porous media, i.e., the shape dependence of the grand canonical potential and of surface energies of a fluid bounded by an arbitrarily shaped convex pore, can be calculated in the thermodynamic limit fully from the knowledge of the Minkowski functionals, i.e., of only four morphometric measures. This remarkable result is based on Hadwiger's theorem on the completeness of the additive Minkowski functionals and the assumption that a thermodynamic potential is an 'additive' functional which can be understood as a more precise definition for the conventional term 'extensive'. As a consequence, the surface energy and other thermodynamic quantities contain in the thermodynamic limit, beside a constant term, only contributions linear in the mean and Gaussian curvature of the pore and not an infinite number of curvature terms. Finally, starting from a microscopic density functional for an inhomogeneous fluid in a porous medium the phase coexistence (capillary condensation) and the critical point of the fluid is determined in terms of structure functions and morphological measures of the pore space and calculated explicitly for specific random porous structures using results from integral geometry.

In a previous paper the author has generalized the Kahler angle to the multiple Kahler angle and formulated a Poincare formula for any real submanifolds in complex projective spaces ℂPn using the multiple Kahler angles of the submanifolds. In this paper we formulate a Poincare formula for submanifolds M and N of complementary dimension in ℂPn by a symmetric polynomial of degree one in cos2θi and cos2τj, where θi and τj are the multiple Kahler angles of M and N. We also obtain some inequalities between the integral of the intersection numbers of submanifolds and their volumes in some cases.

Topography and composition images of model thin films of deuterated polystyrene (dPS) and polyisoprene with different blend compositions were analyzed with an extension of integral-geometry approach. Surface patterns, formed in the course of spin-casting from toluene onto self-assembled monolayers (SAM), were recorded with scanning force microscopy. Their relation with lateral phase domain structures was demonstrated by dynamic secondary ion mass spectrometry, yielding maps of dPS distribution. Morphological measures, which cannot be provided by Fourier transform analysis (FTA), characterize individual images, compositional series of the surface patterns and individual features of the patterns. Different morphologies (nucleation- and spinodal-type and hole- and island-dominated ones) are consistently characterized by the Minkowski measures and related parameters. For instance, the latter can measure circular character of the individual features and estimate dominant lateral length (determined rigorously w...

This work consists of three parts, the first and second of which are concerned with generalized forms of two conjectures by Wm. J. Firey(1978). Let K, L ⊂ R n be compact convex sets which have common interior points and intersect almost transversally. Let Kr ,L r, r ≥ 0, denote the outer parallel bodies of K, L at distance r and set Sr := ∂Kr ∩ ∂Lr, Hr := ∂Kr ∩ Lr .I t has been conjectured by Firey that the Euler characteristic of these sets is independent of r> 0. More generally, it has been shown in (10) that Sr and S0 as well as Hr and H0 are homotopy equivalent for all r ≥ 0. In the present work, we prove that, for all r ≥ 0, Sr and S0 as well as Hr and H0 are in fact bi-lipschitz homeomorphic lipschitz submanifolds of R n . In the second part, we establish translative integral geometric formulae involving such intersections for arbitrary pairs of compact convex sets. Formulae of this type have recently been proved in (10) for compact convex sets with interior points, while very special cases of these have been conjectured by Firey. Finally, the last part is devoted to a theoretical study of general convex surfaces in stochastic geometry.

Very efficient and unbiased principles exist for estimating the total three-dimensional (3D), two-dimensional and zero-dimensional amounts of arbitrary structure in 3D space. The total one-dimensional length of real structure, in the ordinary sense, is an abstraction from the point of view of integral geometry. All stereological estimators of 'tubular length' are thus approximations. In addition, they are riddled by biases due to several types of artificial edges and other practical problems. This paper discusses several of these and proposes practical solutions of minimal biases.

SUMMARY The general objective of this communication is to present the basics of mathematical texture analysis as part of integral geometry involving spherical analogues of the X-ray and Radon transform, and in particular to clarify its relationship with mathematical tomography. Copyright ? 2002 John Wiley & Sons, Ltd.

The theory of valuations on convex sets is a classical part of the topic of onvexity, with traditionally strong relations to integral geometry. During the roughly last 15 years a considerable progress was made in valuation theory and its applications to integral geometry.

Let K,L IR n be two convex bodies with non-empty interiors and with boundaries @K, @L, and let denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t 7! (@K \ (@L + t)) and t 7! (@K \ (L + t)), t 2 IR n , with respect to an n- dimensional Haar measure of IR n can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let Kr,Lr denote the outer parallel bodies of K,L at distance r 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of @Kr \ @Lr and @Kr \ Lr, respectively, is independent of r 0 if K \ L 6 ; and if @K and @L intersect almost transversally. As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey.

Curvature measures are important for the characterization of spatial structures since many physical phenomena depend essentially on the geometry of spatial configurations. Curvature-weighted correlation functions can be defined and calculated explicitly within the Boolean model. This standard model in statistical physics generates random geometries by overlapping grains (spheres, sticks) each with arbitrary location and orientation. A general decomposition relation for characteristic functions of submanifolds based on Chern's differential kinematic formula is used to obtain exact expressions for mean values and second order moments of curvature integrals, i.e., of Minkowski functionals. An exact relation for the canonical and grand-canonical second order moments of the morphological Minkowski measures can be derived based only on thermodynamic arguments. The results are applied on the morphological thermodynamics of complex fluids where specific heats are expressed in terms of curvature moments.

If κ denotes the curvature and τ the torsion of a closed, generic, and oriented polygonal space curve X in \(\mathbb{E}^3\), then we show that ∫X √(κ2 + τ2) ds = ∫X κ ds + ∫X | τ | ds > 4π if τ is positive. We also show that ∫X √(κ2 + τ2) ds ≥ 2πn if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.