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We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers. [ABSTRACT FROM AUTHOR]

Mixed volume is an important notion in convex geometry, which is the extension of volume and surface area. The Minkowski inequality for mixed volume plays a vital role in convex geometry. This paper obtains that mixed volume under Steiner symmetrization is monotonic and decreasing, and a concise proof of the general Minkowski inequality by Steiner symmetrization is obtained. [ABSTRACT FROM AUTHOR]

Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length, defined as a line integral of the condition number along all the lifted renormalized paths. The theory developed in this paper leads to a continuation algorithm tracking all the solutions between two generic systems with the same structure. The algorithm is randomized, in the sense that it follows a random path between the two systems. The probability of success is one. In order to produce an expected cost bound, several invariants depending solely on the supports of the equations are introduced. For instance, the mixed area is a quermassintegral that generalizes surface area in the same way that mixed volume generalizes ordinary volume. The facet gap measures for each 1-cone in the fan and for each support polytope, how close is the supporting hyperplane to the nearest vertex. Once the supports are fixed, the expected cost depends on the input coefficients solely through two invariants: the renormalized toric condition number and the imbalance of the absolute values of the coefficients. This leads to a nonuniform polynomial complexity bound for polynomial solving in terms of those two invariants. [ABSTRACT FROM AUTHOR]

We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ\Gamma in a Cartan-Hadamard manifold MM. In particular, we show that the first mean curvature integral of a convex hypersurface γ\gamma nested inside Γ\Gamma cannot exceed that of Γ\Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ\Gamma in terms of the volume it bounds in MM in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ\gamma is parallel to Γ\Gamma , or MM has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.

We overview the main ideas and techniques of the functional-analytical approach to some extremal problems of convex geometry that stem from the Queen Dido problem. [ABSTRACT FROM AUTHOR]

qaaa We provide a simple proof for the equality between the normalized volume of a convex polytope with m vertices and the mixed volume of m simplices and thus show the seemingly restrictive problem of computing the mixed volume of simplices is still at least as hard as computing volumes of convex polytopes in general. [ABSTRACT FROM AUTHOR]

In the paper, our main aim is to introduce a new concept and call it the - mixed geominimal surface area of convex bodies , which obeys the classical basic properties. The new affine geometric quantity in a special case yields Petty's geominimal surface area of a convex body , Lutwak's -geominimal surface area of , and the newly established mixed geominimal surface area of convex bodies . We establish some -mixed geominimal surface area inequalities for the -mixed geominimal surface area, whose some special cases are Petty's geominimal surface area inequality, Lutwak's -geominimal surface area inequality, and some new mixed geominimal surface area inequalities. [ABSTRACT FROM AUTHOR]

We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra. [ABSTRACT FROM AUTHOR]

The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume V (K 1 , K 2 , K 3 , ... , K n) of convex bodies K 1 , ... , K n in ℝ n by the product of the mixed volumes V (K 1 , K 1 , K 3 , ... , K n) and V (K 2 , K 2 , K 3 , ... , K n). As a consequence, for integers α 1 , ... , α m ∈ ℕ with α 1 + ⋯ + α m = n the product V n (K 1) α 1 n ⋯ V n (K m) α m n of suitable powers of the volumes V n (K i) of the convex bodies K i , i = 1 , ... , m , is a lower bound for the mixed volume V (K 1 [ α 1 ] , ... , K m [ α m ]) , where α i is the multiplicity with which K i appears in the mixed volume. It has been conjectured by Betke and Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume V (K 1 [ α 1 ] , ... , K m [ α m ]) in terms of the product of the intrinsic volumes V α i (K i) , for i = 1 , ... , m. The case where m = 2 , α 1 = 1 , α 2 = n − 1 has recently been settled by the present authors (2020). The case where m = 3 , α 1 = α 2 = 1 , α 3 = n − 2 has been treated by Artstein-Avidan et al. under the assumption that K 2 is a zonoid and K 3 is the Euclidean unit ball. The case where α 2 = ⋯ = α m = 1 , K 1 is the unit ball and K 2 , ... , K m are zonoids has been considered by Hug and Schneider. Here, we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov–Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well. [ABSTRACT FROM AUTHOR]

This study is in line with two previous studies by the same authors on gas injection in yield stress fluids. Gas is injected toward the bottom wall of a prismatic tank containing a yield stress fluid. When rising toward the free surface, trains of bubbles generate fluid recirculation in the tank. Two experimental colorimetric methods are introduced and validated in order to quantify the recirculation liquid flow rate as well as the time evolution of the extent and shape of the mixed volume. The influences of the injection flow rate, fluid rheology, and reactor size have been quantified. Correlations based on the characteristic nondimensional numbers of the flow have been developed to predict the downward liquid flow rate as well as the mixed volume. A model for estimating the mixing time is also developed and compared to experimental results. [ABSTRACT FROM AUTHOR]

Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, y = x 1 x 2 x 3 , over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem. [ABSTRACT FROM AUTHOR]

We classify all tuples of lattice polyhedra of relative mixed volume 1 and all minimal (by inclusion) tuples of polyhedra of relative mixed volume 2. We also prove a conjecture by Esterov, which states that all tuples with finite relative mixed volume are contained in one of finitely many ones that are minimal by inclusion. [ABSTRACT FROM AUTHOR]

We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume m in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals. [ABSTRACT FROM AUTHOR]

This paper concerns the evolution of a closed hypersurface of dimension n (≥ 2) in the Euclidean space ℝ n + 1 under a mixed volume preserving flow. The speed equals a power β (≥ 1) of homogeneous curvature functions of degree one and either convex or concave plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface. [ABSTRACT FROM AUTHOR]

A recent article Li and Lv (J. Geom. Anal. 30: 417–447, 2020) considered contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in certain cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article, we extend the result to various other cases that are analogous to those considered in other earlier work, and we show that in all cases, where sufficient pinching conditions are assumed on the initial hypersurface, then under suitable rescaling the final point is asymptotically round and convergence is exponential in the C ∞ -topology. [ABSTRACT FROM AUTHOR]

Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space R d or on a sphere and other manifolds which in addition satisfy certain edge length constraints. One of the major open problems in this field is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely related to the classification of rigid graphs according to their maximal number of real embeddings. In this paper, we are interested in finding edge lengths that can maximize the number of real embeddings of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain embeddings in R 3. One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration. Our results include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in R 2 and R 3 , while in the case of S 2 we achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum number of realizations. The methods and the results concerning the spatial embeddings are part of the proceedings of ISSAC 2018 (Bartzos et al., 2018). [ABSTRACT FROM AUTHOR]

Rigid graph theory is an active area with many open problems, especially regarding embeddings in R d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C d and S d . The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5 , both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S 2 and C 3 . We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall. [ABSTRACT FROM AUTHOR]

The technical (practical) stability problem for a set of trajectories of discrete systems on a metric space of nonempty convex compact sets in is considered. On the basis of known results of convex geometry and comparison method, an approach of constructing the auxiliary Lyapunov functionals for the study of technical stability in terms of two measures of evolutionary equations with Hukuhara difference operator is proposed. The problem of estimating the solutions of equations is reduced to the study of finite-dimensional difference equations of comparison. Examples of technical stability study are given to illustrate the constructiveness of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we first propose and establish a new Aleksandrov–Fenchel type inequality involving logarithms by introducing two new concepts of mixed volume measure and L p -multiple mixed volume measure, and using the L p -Aleksandrov–Fenchel inequality for the L p -multiple mixed volumes. The new Aleksandrov–Fenchel inequality involving logarithms in special case yields the classical Aleksandrov–Fenchel inequality, and four recent logarithmic Minkowski inequalities, which are logarithmic Minkowski inequality for mixed volumes, L p -mixed volumes, quermassintegrals and p-mixed quermassintegrals, respectively. Moreover, the logarithmic Aleksandrov–Fenchel inequality is also derived. [ABSTRACT FROM AUTHOR]

The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over previous years. It is well known that a lattice polytope has normalized volume one if and only if its degree is zero. Recently, Esterov and Gusev gave a complete classification result for families of n lattice polytopes in R n whose mixed volume equals one. Here, we give a reformulation of their result involving the novel notion of mixed degree that generalizes the degree similar to how the mixed volume generalizes the volume. We discuss and motivate this terminology, also from an algebro-geometric viewpoint, and explain why it extends a previous definition of Soprunov. We also remark how a recent combinatorial result due to Bihan solves a related problem posed by Soprunov. [ABSTRACT FROM AUTHOR]

Two closely connected extremal problems in convex geometry: the minimal L p surface area and the maximal L p John ellipsoid of a convex body under translation transforms, are solved. It is shown that there exists a unique interior point of a convex body such that they attain their extrema. [ABSTRACT FROM AUTHOR]

Let P1, ..., Pn and Q1, ..., Qn be convex polytopes in ℝn with Pi ⊆ Qi. It is well-known that the mixed volume is monotone: V(P1, ..., Pn) ≤ V(Q1, ..., Qn). We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1, ..., Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 ∪ ... ∪ Pn. In addition, we obtain an analog of Cramer's rule for sparse polynomial systems. [ABSTRACT FROM AUTHOR]

We prove new upper bounds for the degrees in Hilbert's Nullstellensatz and for the Noether exponent of polynomial ideals in terms of the monomial structure of the polynomials involved. Our bounds improve the previously known bounds in the sparse setting and are the first to take into account the different supports of the polynomials. [ABSTRACT FROM AUTHOR]

This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust—this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network—namely, the fully processive dual-site network—which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization. [ABSTRACT FROM AUTHOR]

Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized volume of the convex hull of their union. Under these conditions the problem of computing mixed volume of several polytopes can be transformed into a volume computation problem for a single polytope in the same dimension. We demonstrate through problems from real world applications that substantial reduction in computational costs can be achieved via this transformation in situations where the convex hull of the union of the polytopes has less complex geometry than the original polytopes. We also discuss the important implications of this result in the polyhedral homotopy method for solving polynomial systems. [ABSTRACT FROM AUTHOR]

A new family of ellipsoids associated with a pair of convex bodies in Rn is introduced, which extends the classical John ellipsoids and the evolved Lp John ellipsoids in a distinctiveway. Analogues of John's inclusion and Ball's volume-ratio inequality are established for these new ellipsoids. [ABSTRACT FROM AUTHOR]

In Weil (2001) formulae were proved for stationary Boolean models Z in ℝ d with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅ 0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker's representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case. [ABSTRACT FROM AUTHOR]

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables. [ABSTRACT FROM AUTHOR]

Let X be an n-dimensional manifold and V1,...,Vn⊂C∞(X,R) finite-dimensional vector spaces with Euclidean metric. We assign to each Vi a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle to X. We prove that the average number of isolated common zeros of f1∈V1,...,fn∈Vn is equal to the mixed symplectic volume of these Finsler ellipsoids. If X is a homogeneous space of a compact Lie group and all vector spaces Vi together with their Euclidean metrics are invariant, then the average numbers of zeros satisfy the inequalities, similar to Hodge inequalities for intersection numbers of divisors on a projective variety. This is applied to the eigenspaces of Laplace operator of an invariant Riemannian metric. The proofs are based on a construction of the ring of normal densities on X, an analogue of the ring of differential forms. In particular, this construction is used to carry over the Crofton formula to the product of spheres. [ABSTRACT FROM AUTHOR]

This paper is devoted to the instability of Set Differential Equations (SDEs). Using the geometric inequalities of Brunn–Minkowski and A.D. Aleksandrov, we propose new methods for constructing Lyapunov functions. In combination with the known methods of stability theory, the Lyapunov's direct method, the comparison method and the vector-function method, we establish conditions for the collapse of the solutions of the SDEs. Estimates of the collapse time of solutions are also obtained. Examples of SDEs in spaces of dimension 2 and 3 illustrating general theorems are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new measure of asymmetry, called log-Minkowski measure of asymmetry for planar convex bodies in terms of the L0-mixed volume, and show that triangles are the most asymmetric planar convex bodies in the sense of this measure of asymmetry. [ABSTRACT FROM AUTHOR]

Let C be a closed convex cone in R n , pointed and with interior points. We consider sets of the form A = C ∖ K , where K ⊂ C is a closed convex set. If A has finite volume (Lebesgue measure), then A is called a C -coconvex set, and K is called C -close. The family of C -coconvex sets is closed under the addition ⊕ defined by C ∖ ( A 1 ⊕ A 2 ) = ( C ∖ A 1 ) + ( C ∖ A 2 ) . We develop first steps of a Brunn–Minkowski theory for C -coconvex sets, which relates this addition to the notion of volume. In particular, we establish the equality condition for a Brunn–Minkowski type inequality (with reversed inequality sign) and introduce mixed volumes and their integral representations. For C -close sets, surface area measures and cone-volume measures can be defined, in analogy to similar notions for convex bodies. They are Borel measures on the intersection of the unit sphere with the interior of the polar cone of C . We prove a Minkowski-type uniqueness theorem for C -close sets with equal surface area measures. Concerning Minkowski-type existence problems, we give conditions for a Borel measure to be either the surface area measure or the cone-volume measure of a C -close set. These conditions are sufficient in the first case, and necessary and sufficient in the second case. [ABSTRACT FROM AUTHOR]