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The Busemann–Petty problem of arbitrary measure for symmetric star bodies is proposed and studied by Zvavitch, which is a generalization of the classical Busemann–Petty problem. In this paper, we study the Busemann–Petty-type problem for homogeneous measure for general star bodies. [ABSTRACT FROM AUTHOR]

This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems. [ABSTRACT FROM AUTHOR]

Apollonius defined the circle as the set of points that have a given ratio μ of distances from two given points, where the ratio is not equal to one. In a more general sense, consider two 0-symmetric, bounded, convex bodies K and K ′ , which define two norms. Their unit balls are K and K ′ . The surface of Apollonius is defined as the set of points equidistant from the centres of bodies K and K ′ with respect to the aforementioned norms. In this paper we demonstrate that the surface of Apollonius of two ellipsoids is a quadratic surface. We also examine the circumstances under which this surface becomes a sphere. [ABSTRACT FROM AUTHOR]

Let K be an n-dimensional equiangular cone in a Euclidean vector space. Let 0 < ϕ < π be the common angle between any pair of vectors taken among the generators of the cone. We obtain an explicit formula for the maximum angle of K in terms of n and ϕ . We explain also how to identify a pair { u , v } of unit vectors in K achieving the maximum angle. Finally, we analyze a counterintuitive phenomenon that occurs when n is large: the equiangular cone K can be constructed so that u and v are almost opposite, whereas ϕ is near π / 2 . [ABSTRACT FROM AUTHOR]

Given a real number q and a star body in the n-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The corresponding generalized dual Minkowski problem is studied in this paper. By using variational methods, we solve the generalized dual Minkowski problem for q < 0 , and the even generalized dual Minkowski problem for 0 ≤ q ≤ 1 . We also obtain a sufficient condition for the existence of solutions to the even generalized dual Minkowski problem for 1 < q < n . [ABSTRACT FROM AUTHOR]

The L p chord Minkowski problem based on chord measures and L p chord measures introduced firstly by Lutwak et al. (Comm Pure Appl Math 1–54, 2023) is a very important and meaningful geometric measure problem in the L p Brunn–Minkowski theory. Xi et al. (Adv Nonlinear Stud 23:20220041, 2023) using variational methods gave a measure solution when p > 1 and 0 < p < 1 in the symmetric case. Recently, Guo et al. (Math Ann 2023. 10.1007/s00208-023-02721-8) also obtained a measure solution for 0 ≤ p < 1 by similar methods without the symmetric assumption. In the present paper, we investigate and confirm the orlicz chord Minkowski problem, which generalizes the L p chord Minkowski problem by replacing p with a fixed continuous function φ : (0 , ∞) → (0 , ∞) , and achieve the existence of smooth solutions to the orlicz chord Minkowski problem by using methods of Gauss curvature flows. [ABSTRACT FROM AUTHOR]

We approximate boundaries of convex polytopes X ⊂ R n by smooth hypersurfaces Y = Y ε with positive mean curvatures and, by using basic geometric relations between the scalar curvatures of Riemannian manifolds and the mean curvatures of their boundaries, establish lower bound on the dihedral angles of X. [ABSTRACT FROM AUTHOR]

The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d + 1) (r - 1) + 1 points in R d , one can find a partition X = X 1 ∪ ⋯ ∪ X r of X, such that the convex hulls of the X i , i = 1 , ... , r , all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span ⌊ n / 3 ⌋ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Álvarez-Rebollar et al. guarantees ⌊ n / 6 ⌋ pairwise crossing triangles. Our result generalizes to a result about simplices in R d , d ≥ 2 . [ABSTRACT FROM AUTHOR]

In this paper we present two Fenchel-type dual problems for a DC (difference of convex functions) optimization primal one. They have been built by means of the c-conjugation scheme, a pattern of conjugation which has been shown to be suitable for evenly convex functions. We study characterizations of weak, strong and stable strong duality for both pairs of primal–dual problems. We also give conditions which relate the existence of strong and stable strong duality for both pairs. [ABSTRACT FROM AUTHOR]

Characterizations of finite sequences β 1 < ⋯ < β n representing expected values of order statistics from a random sample of size n are given. As a by-product, a characterization of binomial mixtures, when the mixing random variable is supported in the open interval (0, 1), is presented; this enables the exact description of the convex hull of the open binomial curve, as well as the open moment curve. [ABSTRACT FROM AUTHOR]

The L p dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the L p dual Minkowski problem, asks about existence of a convex body with prescribed L p dual curvature measure. This question unifies the previously disjoint L p Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the L p dual Minkowski problem for the case of q < p + 1 , - 1 < p < 0 , and p ≠ q for even measures. [ABSTRACT FROM AUTHOR]

The sub-defect of a doubly stochastic matrix AA, denoted as sd(A)=⌈n−sum(A)⌉sd\left(A)=\lceil n-{\rm{sum}}\left(A)\rceil , is defined as the minimum number of rows and columns required to be added to transform the doubly substochastic matrix into a doubly stochastic matrix. Here, nn signifies the matrix size, and sum(A){\rm{sum}}\left(A) represents the sum of all entries of AA. This article systematically examines the sub-defect characteristics inherited in doubly stochastic matrices, specifically in the context of symmetric, Hankel-symmetric, and centrosymmetric doubly substochastic matrices. Furthermore, we present illustrative examples to elucidate the practical applicability and significance of our approach in comprehending and manipulating the sub-defect of these specialized matrices.

In this paper, we introduce the L p q-torsional measure for p ∈ ℝ and q > 1 by the L p variational formula for the q-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the L p Minkowski problem with respect to the q-torsional rigidity for discrete measures and general measures when 0 < p < 1 and q > 1 . [ABSTRACT FROM AUTHOR]

In this paper, we deal with the torsion log-Minkowski problem without symmetry assumptions via an approximation argument. [ABSTRACT FROM AUTHOR]

We prove that if u : R n → R is strongly convex, then for every ε > 0 there is a strongly convex function v ∈ C 2 (R n) such that | { u ≠ v } | < ε and ∥ u − v ∥ ∞ < ε . [ABSTRACT FROM AUTHOR]

We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. ), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces. [ABSTRACT FROM AUTHOR]

For any Wulff shape W, its dual Wulff shape and spherical Wulff shape W ~ can be defined naturally. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, we prove that a polytope is self-dual if and only if its spherical Wulff shape is a spherical convex body of constant width. We also prove that a smooth Wulff shape is self-dual if and only if for any interior points P of W ~ and for any point Q of the intersection of the boundary of W ~ and the graph of its spherical support function (with respect to P), the image of Q under the spherical blow-up (with respect to P) is always a point of W ~ . Moreover, we give an affirmative answer to the problem posed by M. Lassak which says that “Do there exist reduced spherical n-dimensional polytopes (possibly some simplices?) on S n , where n ≥ 3 , different from the 1 / 2 n part of S n ? ”. [ABSTRACT FROM AUTHOR]

In this paper, we focus on covering and illumination properties of a specific class of convex polytopes denoted by P . These polytopes are obtained as the convex hull of the Minkowski sum of a finite subset of Z n and (1 / 2) [ - 1 , 1 ] n . Our investigation includes the verification of Hadwiger's covering conjecture for P , as well as the estimation of the covering functional for convex polytopes in P . Furthermore, we demonstrate that when an integer M is sufficiently large, the elements belonging to P that are contained in M [ - 1 , 1 ] n serve as an ε -net for the space of convex bodies in R n , equipped with the Banach–Mazur metric. [ABSTRACT FROM AUTHOR]

We provide a characterization of those log-concave distributions in R n that are contoured distributions, through the K p -bodies of the distribution, defined by Ball. Our method uses the logarithmic integral for the solution of a Bernstein type approximation problem. In the second part of the paper we state a question for contoured distributions that would provide an alternative approach to the isotropic constant problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an m-fold illumination number I m (K) of a convex body K in Euclidean space E d , which is the smallest number of directions required to m-fold illuminate K, i.e., each point on the boundary of K is illuminated by at least m directions. We get a lower bound of I m (K) for any d-dimensional convex body K, and get an upper bound of I m (K) for any d-dimensional convex body K with smooth boundary. We also prove that I m (K) = 2 m + 1 , for a 2-dimensional smooth convex body K. Furthermore, we obtain some results related to the m-fold illumination numbers of convex polygons and cap bodies of a d-dimensional unit ball B d in small dimensions. In particular, we show that , for a regular convex n-sided polygon P. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 and let K ⊂ ℝn be a convex body. A point p ∈ int K is said to be a Larman point of K if for every hyperplane Π passing through p, the section Π ∩ K has an (n – 2)-plane of symmetry. If p is a Larman point of K and for every section Π ∩ K, p is in the corresponding (n – 2)-plane of symmetry, then we call p a revolution point of K. We conjecture that if K contains a Larman point which is not a revolution point, then K is either an ellipsoid or a body of revolution. This generalizes a conjecture of Bezdek for n = 3. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if K ⊂ ℝn is a strictly convex origin symmetric body that contains a revolution point p which is not the origin, then K is a body of revolution. This generalizes the False Axis of Revolution Theorem in [7]. We also show that if p is a Larman point of K ⊂ ℝ3 and there exists a line L such that p ∉ L and, for every plane Π passing through p, the line of symmetry of the section Π ∩ K intersects L, then K is a body of revolution (in some cases, K is a sphere). We obtain a similar result for projections of K. Additionally, for K ⊂ ℝn with n ≥ 4, we show that if every hyperplane section or projection of K is a body of revolution and K has a unique diameter D, then K is a body of revolution with axis D. [ABSTRACT FROM AUTHOR]

Using the connection between ellipsoids and positive semidefinite matrices we provide alternative proofs to some recently proven inequalities concerning the volume of L 2 zonoids as consequences of classical inequalities for matrices. [ABSTRACT FROM AUTHOR]

Three hundred years ago, Prince Rupert of Rhine showed that a unit cube has the property that one copy of it can be passed through a suitable hole in another copy. Under this situation, we say that a unit cube has the Rupert property. In the past years, there are many research studying about the Rupert property of many solids in R 3 . For higher dimensions, the n-dimensional cube and the regular n-simplex were studied to have the Rupert property. In this work, we focus on the Rupert property of some polyhedrons in n dimensions. In particular, we show that some particular n-dimensional simplices, generalized n-dimensional octahedrons and some related solids in R n have the Rupert property using arbitrarily small rotations and translations. [ABSTRACT FROM AUTHOR]

We study the open, closed, and non-degenerate embedding dimensions of neural codes, which are the smallest respective dimensions in which one can find a realization of a code consisting of convex sets that are open, closed, or non-degenerate in a sense defined by Cruz, Giusti, Itskov, and Kronholm. For a given code C we define the embedding dimension vector to be the triple (a, b, c) consisting of these embedding dimensions. Existing results guarantee that max { a , b } ≤ c , and we show that when any of these dimensions is at least 2 this is the only restriction on such vectors. Specifically, for every triple (a, b, c) with 2 ≤ min { a , b } and max { a , b } ≤ c ≤ ∞ we construct a code C (a , b , c) whose embedding dimension vector is exactly (a, b, c) (where an embedding dimension is ∞ if there is no realization of the corresponding type). Our constructions combine two existing tools in the convex neural codes literature: sunflowers of convex open sets, and rigid structures, the latter of which was recently defined in work of Chan, Johnston, Lent, Ruys de Perez, and Shiu. Our constructions provide the first examples of codes whose closed embedding dimension is larger than their open embedding dimension, but still finite. [ABSTRACT FROM AUTHOR]

Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or 'nice') which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone. [ABSTRACT FROM AUTHOR]

In this paper, we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in R n + 1 with speed ψ σ k α (λ) , where α > 1 k is a constant, σ k (λ) is the k-th elementary symmetric polynomial of the principal radii of curvature and ψ is a preassigned positive smooth function defined on S n . We prove that under some assumptions of ψ , the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known L p Christoffel-Minkowski problem u 1 - p (x) σ k (∇ 2 u + u I) = c ψ (x) for 1 < p < k + 1 . [ABSTRACT FROM AUTHOR]

This paper aims to establish a basic framework for the dual Brunn–Minkowski theory for unbounded closed convex sets in C, where C ⊊ R n is a pointed closed convex cone with nonempty interior. In particular, we provide a detailed study of the copolarity, define the C-compatible sets, and establish the bipolar theorem related to the copolarity of the C-compatible sets. Based on these notions, we are able to define the q-th dual curvature measures for C-compatible sets, and propose its corresponding dual Minkowski problem. For the C-determined sets, this dual Minkowski problem is solved for all 0 ≠ q ∈ R . For q > 0 , we also establish the existence of solutions to the dual Minkowski problem for (C, q)-close sets and hence provide a weak solution to a Monge–Ampère type equation. The case for q = 0 is discussed as well which naturally leads to the Alexandrov problem for C-compatible sets. [ABSTRACT FROM AUTHOR]

In this paper, we propose new proofs of the classical Brunn—Minkowski theorem on the volume of the sum of convex polyhedra P0 and P1 of the same n-dimensional volume in Euclidean space ℝn, n ≥ 2: Vn((1 − t)P0 + tP1) ≥ Vn(P0) = Vn(P1), 0 < t < 1, where the equality holds only if P1 is obtained from P0 by a parallel translation; in other cases, the strict inequality holds. Proofs are based on the sequential partition of the polyhedron P0 into simplexes by hyperplanes. For dimensions n = 2 and n = 3, in the case where P0 is a simplex (a triangle for n = 2), for an arbitrary convex polyhedron P1 ⊂ ℝn, we construct a continuous (in the Hausdorff metric) one-parameter family of convex polyhedra P1(s) ⊂ ℝn, s ∈ [0, 1], P1(0) = P1, for which the function w(s) = Vn((1 − t)P0 + tP1(s)) strictly monotonically decreases, and P1(1) is obtained from P0 by a parallel translation. If P1 is not obtained from P0 by a parallel translation, then, using elementary geometric constructions, we establish the existence of a polyhedron P 1 ′ for which Vn ((1 − t)P0 + tP1) > Vn((1 − t)P0 + t P 1 ′ ). [ABSTRACT FROM AUTHOR]