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We show that a compact Riemannian 3-manifold M with strictly convex simply connected boundary and sectional curvature K\leq a\leq 0 is isometric to a convex domain in a complete simply connected space of constant curvature a, provided that K\equiv a on planes tangent to the boundary of M. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard 3-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for K\geq a\geq 0. [ABSTRACT FROM AUTHOR]

For each natural number n and any bounded, convex domain \Omega \subset \mathbb {R}^n we characterize the sharp constant C(n,\Omega) in the Poincaré inequality \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})}. Here, \bar {f}_{\Omega } denotes the mean value of f over \Omega. In the case that \Omega is a ball B_r of radius r in \mathbb {R}^n, we calculate C(n,B_r)=C(n)r explicitly in terms of n and a ratio of the volumes of the unit balls in \mathbb {R}^{2n-1} and \mathbb {R}^n. More generally, we prove that C(n,B_{r(\Omega)}) \leq C(n,\Omega) \leq \frac {n}{n+1}\mathrm {diam}(\Omega), where B_{r(\Omega)} is a ball in \mathbb {R}^n with the same n-dimensional Lebesgue measure as \Omega. Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant. [ABSTRACT FROM AUTHOR]

In this work, we establish precise local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains \Omega \subset \mathbb {R}^3 with smooth boundary \partial \Omega \neq \emptyset. This result is the improved estimates established by Len Meas [C. R. Math. Acad. Sci. Paris 355 (2017), pp. 161–165]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain \Omega have been proved by Blair, Smith, Sogge [Proc. Amer. Math. Soc. 136 (2008), pp. 247–256; Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), pp.1817–1829]. Optimal estimates in strictly convex domains have been obtained by Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380]. Our case of cylindrical domains is an extension of the result of Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions. [ABSTRACT FROM AUTHOR]

We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of a half-space we obtain the sharp constant while for a general convex domain we obtain estimates that are better than those obtained by comparison with the polyharmonic operator. [ABSTRACT FROM AUTHOR]

We prove a spectral upper bound for the torsion function of symmetric stable processes that holds for convex domains in \mathbb {R}^d. Our bound is explicit and captures the correct order of growth in d, improving upon the existing results of Giorgi and Smits [Indiana Univ. Math. J. 59 (2010), pp. 987–1011] and Biswas and Lőrinczi [J. Differential Equations 267 (2019), pp. 267–306]. Along the way, we make progress towards a torsion analogue of Chen and Song's [J. Funct. Anal. 226 (2005), pp. 90–113] two-sided eigenvalue estimates for subordinate Brownian motion. [ABSTRACT FROM AUTHOR]

We investigate some monotonicity properties of the best constant from Morrey's inequality in convex and bounded domains from the Euclidean space \mathbb R^D (D\geq 1). Using these monotonicity properties we give a new variational characterization of the best constant from Morrey's inequality on convex and bounded domains for which the maximum of the distance function to the boundary is small. We also show that this variational characterization does not hold true on convex and bounded domains for which the maximum of the distance function to the boundary is larger than 1. [ABSTRACT FROM AUTHOR]

We present an iterative method based on repeatedly inverting the Monge-Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain Ω ⊂ Rn. We prove that the iterates uk generated by this method converge as k → ∞ to a solution of the Monge-Ampère eigenvalue problem {detD2u = λMA (−u)n in Ω, u = 0 on ∂ Ω. Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates uk := uk/||uk||L∞(Ω) converge to the eigenfunction of unit height. In addition, we show that limk → ∞ R(uk) = limk → ∞ R(uk) = λMA, where the Rayleigh quotient R(u) is defined as R(u) := ∫Ω (−u) det D2u/∫Ω (−u)n+1. Our method converges for a wide class of initial choices u0 that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Ampère eigenvalue λMA. [ABSTRACT FROM AUTHOR]

We provide a sufficient condition for the continuous extension of isometries for the Kobayashi distance between bounded convex domains in complex Euclidean spaces having boundaries that are only slightly more regular than C1. This is a generalization of a recent result by A. Zimmer. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ Rn be a convex domain, and let ƒ:Ω → R be a subharmonic function, Δ ƒ ≥ 0, which satisfies ƒ ≥ 0 on the boundary ∂ Ω. Then ∫Ω ƒ dx ≤ |Ω|1/n ∫∂ Ω ƒ d σ. Our proof is based on a new gradient estimate for the torsion function, Δ u = −1 with Dirichlet boundary conditions, which is of independent interest. [ABSTRACT FROM AUTHOR]

Let 1 ≤ q ≤ (n−1). We first show that a necessary condition for a Hankel operator on (0,q−1)-forms on a convex domain to be compact is that its symbol is holomorphic along q-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension (n−1) (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level q if and only if they hold at all levels. [ABSTRACT FROM AUTHOR]

In [Invent. Math. 207 (2017), pp. 1289-1299] it is claimed that Kobayashi isometries in strictly convex domains are holomorphic or antiholomorphic. Generally, following the idea of St. Antonakoudis of using the relation between Klein and PoincarÉ models, we give simplified and complete proof of this fact. [ABSTRACT FROM AUTHOR]

We establish a new characterization for a conformal mapping of the unit disk D to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets of λ of the Poincaré metric λ|dw| of a convex domain are strictly convex. [ABSTRACT FROM AUTHOR]

Unlike the case in one dimension, there is still much to learn about the basic nature of the family K(B) of normalized (f(0) = 0, Df(0) = I, where Df is the Fréchet derivative of f and I is the identity operator on Cn) biholomorphic mappings f of the Euclidean unit ball B ⊆ Cn onto convex domains in Cn when n ≥ 2. We consider its closed convex hull coK(B) in relation to the family R(B) of normalized holomorphic mappings f : B → Cn satisfying Re〈f(z), z〉 > ‖z‖²/2 for z ∈ B \{0}, where 〈·,·〈 and ‖·‖ are, respectively, the Hermitian inner product and Euclidean norm in Cn. In dimension n = 1, the sets are the same. Here, we identify some extreme points of coK(B) and use them to show that coK(B) is a proper subset of R(B) when n ≥ 2. We also consider an extension operator related to R(B) that helps illustrate where the known extreme points of coK(B) lie in R(B) and make some observations on the related case of the unit polydisk. [ABSTRACT FROM AUTHOR]

Unlike the case in one dimension, there is still much to learn about the basic nature of the family K(B) of normalized (f(0) = 0, Df(0) = I, where Df is the Fréchet derivative of f and I is the identity operator on Cn) biholomorphic mappings f of the Euclidean unit ball B ⊆ Cn onto convex domains in Cn when n ≥ 2. We consider its closed convex hull coK(B) in relation to the family R(B) of normalized holomorphic mappings f : B → Cn satisfying Re〈f(z), z〉 > ‖z‖²/2 for z ∈ B \{0}, where 〈·,·〈 and ‖·‖ are, respectively, the Hermitian inner product and Euclidean norm in Cn. In dimension n = 1, the sets are the same. Here, we identify some extreme points of coK(B) and use them to show that coK(B) is a proper subset of R(B) when n ≥ 2. We also consider an extension operator related to R(B) that helps illustrate where the known extreme points of coK(B) lie in R(B) and make some observations on the related case of the unit polydisk. [ABSTRACT FROM AUTHOR]

We show that every limit point of a Zariski dense discrete subgroup Γ of the isometry group of a symmetric space of noncompact type is conical if and only if Γ is convex cocompact. [ABSTRACT FROM AUTHOR]

In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton-Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton-Jacobi equations is also obtained. [ABSTRACT FROM AUTHOR]

We present a geometric and elementary proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below. [ABSTRACT FROM AUTHOR]

Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, and the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non-simply connected domains, generalizing a recent example of Fornæss. [ABSTRACT FROM AUTHOR]

For a pseudoconvex tube domain, we prove estimates that relate the sublevel sets of its diagonal Bergman kernel to the floating bodies of its convex base. This allows us to associate a new affine invariant to any convex body. [ABSTRACT FROM AUTHOR]

Let Kv (×) be the modified Bessel functions of the second kind of order v. We prove that the function × ↔∞Ku (×)Kv (×) /K(u+v)/2 ×2 is strictly decreasing on (0,∞). Our study not only involves the Turán type inequalities, log-conve×ity or log-concavity of Kv (×), and the conjecture posed by Baricz, but also yields various new results concerning the monotonicity and conve×ity of the ratios of the modified Bessel functions of the second kind. As applications of our main theorems, some new sharp inequalities involving Kv (×) are presented, which contain sharp estimates for Kv (×) and sharp bounds for the ratios K'v (×)/Kv (×) and Kv=1(×) /Kv (×). [ABSTRACT FROM AUTHOR]

In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in R3 tends to infinity as its genus or the number of boundary components tends to infinity. [ABSTRACT FROM AUTHOR]

We study the Dirichlet eigenvalues of the Laplacian on a convex domain in Rn, with n ≥ 2. In particular, we generalize and improve upper bounds for the Riesz means of order σ ≥ 3/2 established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a negative second term in the Berezin inequality. The obtained remainder term reflects the correct order of growth in the semi-classical limit and depends only on the measure of the boundary of the domain. We emphasize that such an improvement is for general Ω ⊂ Rn not possible and was previously known to hold only for planar convex domains satisfying certain geometric conditions. As a corollary we obtain lower bounds for the individual eigenvalues λk, which for a certain range of k improves the Li-Yau inequality for convex domains. However, for convex domains one can by using different methods obtain even stronger lower bounds for λk. [ABSTRACT FROM AUTHOR]

We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains. [ABSTRACT FROM AUTHOR]

It is proved that any smooth open Whitney umbrella in C² is locally polynomially convex near the singular point. [ABSTRACT FROM AUTHOR]

The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasiconformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric. [ABSTRACT FROM AUTHOR]

We give a generalization of a result of R. Pandharipande to arbitrary characteristic: We prove that, if X is a convex, separably rationally connected, smooth complete intersection in PN over an algebraically closed field of arbitrary characteristic, then X is rational homogeneous. [ABSTRACT FROM AUTHOR]

G. Parisi predicted an important variational formula for the thermodynamic limit of the intensive free energy for a class of mean field spin glasses. In this paper, we present an elementary approach to the study of the Parisi functional using stochastic dynamic programing and semi-linear PDE. We give a derivation of important properties of the Parisi PDE avoiding the use of Ruelle Probability Cascades and Cole-Hopf transformations. As an application, we give a simple proof of the strict convexity of the Parisi functional, which was recently proved by Auffinger and Chen. [ABSTRACT FROM AUTHOR]

We show that convex entire solutions to Donaldson's equation are quadratic, using a result of Weiyong He. We also exhibit entire solutions to the Donaldson equation that are not of the form discussed by He. In the process we discover some nontrivial entire solutions to complex Monge-Ampère equations. [ABSTRACT FROM AUTHOR]

A class of pseudoconvex domains in ℂn generalizing the Hartogs triangle is considered. The Lp boundedness of the Bergman projection associated to these domains is established, for a restricted range of p depending on the "fatness" of domains. This range of p is shown to be sharp. [ABSTRACT FROM AUTHOR]

We prove slicing inequalities for measures of the unit balls of subspaces of Lp, −∞ < p < ∞. For example, for every k ∈ N there exists a constant C(k) such that for every n ∈ N, k < n, every convex k-intersection body (unit ball of a normed subspace of L−k) L in R n and every measure µ with non-negative even continuous density in R n, ... where ξ ⊥ is the central hyperplane in R n perpendicular to ξ, and |L| is the volume of L. This and other results are in the spirit of the hyperplane problem of Bourgain. The proofs are based on stability inequalities for intersection bodies and estimates for the Banach-Mazur distance from the unit ball of a subspace of Lp to the class of intersection bodies. [ABSTRACT FROM AUTHOR]

We obtain the explicit form of the expansion of the Jacobi polynomial P(α,β) n (1 − 2x/β) in terms of the negative powers of β. It is known that the constant term in the expansion coincides with the Laguerre polynomial L(α) n (x). Therefore, the result in the present paper provides the higher terms of the asymptotic expansion as β →∞. The corresponding asymptotic relation between the zeros of Jacobi and Laguerre polynomials is also derived. [ABSTRACT FROM AUTHOR]

We prove slicing inequalities for measures of the unit balls of subspaces of Lp, −∞ < p < ∞. For example, for every k ∈ N there exists a constant C(k) such that for every n ∈ N, k < n, every convex k-intersection body (unit ball of a normed subspace of L−k) L in R n and every measure µ with non-negative even continuous density in R n, ... where ξ ⊥ is the central hyperplane in R n perpendicular to ξ, and |L| is the volume of L. This and other results are in the spirit of the hyperplane problem of Bourgain. The proofs are based on stability inequalities for intersection bodies and estimates for the Banach-Mazur distance from the unit ball of a subspace of Lp to the class of intersection bodies. [ABSTRACT FROM AUTHOR]

In this note, we prove that at a singularity of an (m + 1)-convex mean curvature flow, Andrews' non-collapsing ratio improves as much as is allowed by the example of the shrinking cylinder ℝm x Sn-m. More precisely, we show that for any ε > 0 we have ... ≤ (1 + ε) 1/n-m H wherever the mean curvature H is sufficiently large, where k is the interior ball curvature. When (m + 1) < n, this estimate improves the inscribed radius estimate of Brendle, which was subsequently proved much more directly by Haslhofer-Kleiner by using the powerful new local blow-up method they developed in an earlier work. Our estimate is also based on their local blow-up method, but we do not require the structure theorem for ancient flows, instead making use of the gradient term which appears in the evolution equation of the two-point function which defines the interior and exterior ball curvatures. We also obtain an optimal exterior ball estimate for flows of convex hypersurfaces. [ABSTRACT FROM AUTHOR]

New Bonnesen-type inequalities for simply connected domains on surfaces of constant curvature are proved by using integral formulas. These inequalities are generalizations of known inequalities of convex domains. [ABSTRACT FROM AUTHOR]

The inequalities involving the volume of convex bodies play an important role in convex geometry. In this paper, the Rogers and Shephard inequality for the Orlicz difference body of a convex body in the two-dimensional case is established. [ABSTRACT FROM AUTHOR]

Obstacles K and L in ℝd (d ≥ 2) are considered that are finite disjoint unions of strictly convex domains with C³ boundaries. We show that if K and L have (almost) the same scattering length spectrum, or (almost) the same travelling times, then K = L. [ABSTRACT FROM AUTHOR]

We investigate the convexity problem for the Parisi functional defined on the space of the so-called functional ordered parameters in the Sherrington-Kirkpatrick model. In a recent work of Panchenko, it was proved that this functional is convex along one-sided directions with a probabilistic method. In this paper, we will study this problem with a PDE approach that simplifies the original proof and presents more general results. [ABSTRACT FROM AUTHOR]

A deep result of J. Lewis (1983) shows that the polylogarithms Liɑ(z):= σk=1∞ zk/kɑ map the open unit disk D centered at the origin one-to-one onto convex domains for all ɑ ≥ 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain &Lgr;:= C\[1, ∞), introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that Liɑ(z) maps an arbitrary open disk or half-plane in &Lgr; one-to-one onto a convex domain for every ɑ ≥ 1. [ABSTRACT FROM AUTHOR]

It is a well-known result due to Busemann that the intersection body of an origin-symmetric convex body is also convex. Koldobsky introduced the notion of fc-intersection bodies. We show that the k-intersection body of an origin-symmetric convex body is not necessarily convex if k > 1. [ABSTRACT FROM AUTHOR]

A simplex is the convex hull of n + 1 points in ℝn which form an affine basis. A regular simplex ▵ n is a simplex with sides of the same length. We consider the billiard flow inside a regular simplex of ℝn. We show the existence of two types of periodic trajectories. One has period n + 1 and hits each face once. The other one has period 2n and hits n times one of the faces while hitting any other face once. In both cases we determine the exact coordinates for the points where the trajectory hits the boundary of the simplex. [ABSTRACT FROM AUTHOR]

In this article it is shown that the three conditions on the norm ‖·‖ of a Banach space called "geometric convexity", "balanced" and "doubling" in an earlier work by the authors related to eikonal equations are in fact all equivalent. Moreover, each of them is equivalent to a condition called "Property Γ" by Ganichev and Kalton. A fifth condition, that the second derivative of the function t → ‖x + ty‖ is a doubling measure on [-2, 2] for suitable x, y ∈ X, is also equivalent to the various other properties, and this formulation occupies a central place in the analysis. [ABSTRACT FROM AUTHOR]

Given a quasi-Hermitian semisimple Lie algebra, we describe possible spectra of the sum of two admissible elements from its dual vector space. [ABSTRACT FROM AUTHOR]

A variational formulation of Funk metric defined on a convex set in a Euclidean space is introduced. The new definition provides geometric descriptions of the Finsler metric. Secondly, the variational characterization of the Funk metric is generalized to theWeil-Petersson geometry of Teichmüller spaces. Finally, a comparison between several Funk-type metrics defined on Teichmüller spaces is made. [ABSTRACT FROM AUTHOR]

Consider a measurable space with a finite vector measure. This measure defines a mapping of the s-field into a Euclidean space. According to Lyapunov's convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the same given vector measure is also compact and, if the measure is atomless, it is convex. We further provide a geometrically constructed convex compactum in the Euclidean space that contains this union. The equality of these two sets, which holds for two-dimensional measures, can be violated in higher dimensions. [ABSTRACT FROM AUTHOR]

We provide a counterexample (a strongly convex domain in C³) to a statement about certain extremal bases in the papers of J. D. McNeal in 1992 and 1994 which was used to prove estimates for the Bergman kernel and invariant metrics. We also prove that those estimates in fact remain correct for the same bases (other authors had introduced different bases to fix this). [ABSTRACT FROM AUTHOR]

Given a nonnegative function ..., we define the Takagi type function ... where dZ(x) := dist(x, Z) := inf{|x - k| : k ∈ Z}. The main result of the paper states that if φ(0) = 0 and the mapping x → φ(x)/x is concave, then the Takagi type function Sφ is approximately Jensen convex in the following sense: ... Applications to the theory of approximately convex functions are also given. [ABSTRACT FROM AUTHOR]

We prove that any convex-like structure in the sense of Nate Brown is affinely and isometrically isomorphic to a closed convex subset of a Banach space. This answers an open question of Brown. As an intermediate step, we identify Brown's algebraic axioms as equivalent to certain well-known axioms of abstract convexity. We conclude with a new characterization of convex subsets of Banach spaces. [ABSTRACT FROM AUTHOR]

It is shown that intrinsic volumes of a convex body decrease under linear contractions. [ABSTRACT FROM AUTHOR]

By solving the Skorokhod equation for reflecting diffusion processes on a convex domain, gradient estimates for the associated Neumann semigroup are derived. As applications, functional/Harnack inequalities are established for the Neumann semigroup. When the domain is bounded, the gradient estimates are applied to the study of Riesz transforms and regularity of the inhomogeneous Neumann problems on convex domains. [ABSTRACT FROM AUTHOR]

Given a pure, full-dimensional, locally strongly connected polyhedral complex C, we characterize, by a local codimension-2 condition, polyhedral complexes that coarsen C. The proof of the characterization draws upon a general shortcut for showing that a collection of polyhedra is a polyhedral complex and upon a property of hyperplane arrangements which is equivalent, for Coxeter arrangements, to Tits' solution to the Word Problem. The motivating special case, the case where C is a complete fan, generalizes a result of Morton, Pachter, Shiu, Sturmfels, and Wienand that equates convex rank tests with semigraphoids. We also prove oriented matroid versions of our results, obtaining, as a byproduct, an oriented matroid version of Tietze's convexity theorem. [ABSTRACT FROM AUTHOR]