A generalized localization theorem and geometric inequalities for convex bodies

Estimates for measures of lower dimensional sections of convex bodies

In this paper, we will denote by convex figure a compact convex subset of the n-dimensional Euclidean space ℝn, and by convex body a convex figure with non-empty interior. The principal kinematic formula in integral geometry gives the measure of the set of congruent convex bodies intersecting with a fixed convex body. Specifically, let K, L be two convex bodies in ℝn and G(n) the group of special motions in ℝn. Each element, g: ℝn → ℝn, of G(n) can be represented bywhere b∈ℝn and e is an orthogonal matrix of determinant 1. Let μ be the Haar measure on G(n) normalized as follows: Let μ:ℝn × SO(n) → G(n) be defined by φ(t, e)x = ex + t, xeℝn, where SO(n) is the rotation group of ℝn. If v is the unique invariant probability measure on SO(n), η is the Lebesgue measure on ℝn, then μ is chosen as the pull back measure of η⊗v under φ−1. If Wi(K), Wi(L) are the quermassintegrals of K, L, i= 0, 1,…, n, the principal kinematic formula states thatwhere ωn is the volume of the unit n–ball.

Let ‖·‖ be the Euclidean norm on Rn and γn the (standard) Gaussian measure on Rn with density (2π)−n/2e. It is proved that there is a numerical constant c>0 with the following property: if K is an arbitrary convex body in Rn with γn(K)≥1/2, then to each sequence u1,…,um∈Rn with ‖u1‖,…,‖um‖≤c there correspond signs e1,…,em=±1 such that ∑mi=1eiui∈K. This improves the well-known result obtained by Spencer [Trans. Amer. Math. Soc.289, 679–705 (1985)] for the n-dimensional cube. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 351–360, 1998

A [formula omitted] estimate for measures of hyperplane sections of convex bodies

We consider the problem of sampling a point from an arbitrary distribution π over an arbitrary subset S of an integer hyperrectangle. Neither the distribution π nor the support set S are assumed to be available as explicit mathematical equations, but may only be defined through oracles and, in particular, computer programs. This problem commonly occurs in black-box discrete optimization as well as counting and estimation problems. The generality of this setting and high dimensionality of S precludes the application of conventional random variable generation methods. As a result, we turn to Markov chain Monte Carlo (MCMC) sampling, where we execute an ergodic Markov chain that converges to π so that the distribution of the point delivered after sufficiently many steps can be made arbitrarily close to π. Unfortunately, classical Markov chains, such as the nearest-neighbor random walk or the coordinate direction random walk, fail to converge to π because they can get trapped in isolated regions of the support set. To surmount this difficulty, we propose discrete hit-and-run (DHR), a Markov chain motivated by the hit-and-run algorithm known to be the most efficient method for sampling from log-concave distributions over convex bodies in Rn. We prove that the limiting distribution of DHR is π as desired, thus enabling us to sample approximately from π by delivering the last iterate of a sufficiently large number of iterations of DHR. In addition to this asymptotic analysis, we investigate finite-time behavior of DHR and present a variety of examples where DHR exhibits polynomial performance.

For a convex body K C iR, the kth projection function of K assigns to any k-dimensional linear subspace of RI the k-volume of the or thogonal projection of K to that subspace. Let K and Ko be convex bodies in Rn, and let Ko be centrally symmetric and satisfy a weak regularity assump tion. Let i, j E N be such that 1 < i < j < n-2 with (i, j) :A (1, n-2). Assume that K and Ko have proportional ith projection functions and proportional jth projection functions. Then we show that K and Ko are homothetic. In the particular case where Ko is a Euclidean ball, we thus obtain characteri zations of Euclidean balls as convex bodies having constant i-brightness and constant j-brightness. This special case solves Nakajima's problem in arbitrary dimensions and for general convex bodies for most indices (i, j).

Hyperspaces of smooth convex bodies up to congruence

Extremizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

In the first three sections of this chapter, we will investigate four affine invariant problems referring to convex bodies in Rn. It is shown that these problems are equivalent for compact, convex bodies, whereas they differ from each other in the unbounded case. Among these four problems, the central one is the question for the minimal number of smaller homothets of a convex body M ⊂ Rn which are sufficient to coverM. In addition, the problem of illuminating of the boundary bd M by the smallest number of directions is discussed. A lot of partial results regarding both the problems are known, but for n ≥ 3 the general solutions are still unknown. We give a survey on the contributions up to the recent state.KeywordsBoundary PointConvex BodySupporting LineSmall Positive IntegerOuter NormalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The dual Minkowski problem for symmetric convex bodies

Let and let and be two convex bodies in such that their orthogonal projections and onto any ‐dimensional subspace are directly congruent, that is, there exists a rotation and a vector such that . Assume also that the 2‐dimensional projections of and are pairwise different and they do not have ‐symmetries. Then and are congruent. We also prove an analogous more general result about twice differentiable functions on the unit sphere in .

Curvature measure and surface area measure are the basic notions in the classical differential geometry. They play fundamental roles in the theory of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. The Minkowski problem is the problem of prescribing n-th surface area measure on Sn. The Christoffel problem concerns the prescribing the 1-st surface area measure (e.g., see [1, 14, 17, 6, 19, 7, 3]). The general problem of prescribing surface area measures is called the Christoffel-Minkowski problem, we refer [12] for an updated account. The problem of prescribing 0-th curvature measure is called the Alexandrov problem, which is a counterpart to Minkowski problem. The problem is equivalent to solve a Monge-Ampere type equation on Sn. The existence and uniqueness were obtained by Alexandrov [2]. The regularity of the Alexandrov problem in elliptic case was proved by Pogorelov [18] for n = 2 and by Oliker [16] for higher dimension case. The general regularity results (degenerate case) of the problem were obtained in [9]. The general problem of prescribing (n− k)-th curvature measure for case k ≤ n is an interesting counterpart of the Christoffel-Minkowski problem. It has been discussed in literature (e.g., [20]). Nevertheless, very little is known except for the Alexandrov problem. In this paper, we are concerned with the existence of convex bodies with the prescribed (n− k)-th curvature measure for 1 ≤ k < n. We start with the definitions of curvature measures and surface area measures for convex bodies with smooth boundary. Let Ω be a bounded convex body in Rn+1 with C2 boundary M , the corresponding curvature measures and surface area measures of Ω can be defined according to some geometric quantities of M . Let κ = (κ1, · · · , κn) be the principal curvatures of M at point x, let Wk(x) = Sk(κ(x)) be the k-th Weingarten curvature of M at x (where Sk is the k-th elementary symmetric function). In particular, W1,W2

The kth projection function of a convex body K ⊂ ℝn assigns to any k-dimensional linear subspace of ℝn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in ℝn, and let K0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K0 with ∂K0 of class C2 with positive radii of curvature). Assume that K and K0 have proportional 1st projection functions (i.e., width functions) and proportional kth projection functions. For 2 ≤ k < (n + 1)/2 and for k = 3, n = 5, it is shown that K and K0 are homothetic. In the special case where K0 is a Euclidean ball, characterizations of Euclidean balls as convex bodies of constant width and constant k-brightness are thus obtained.

Comparing the volume of the projection body of a double cone and that of the projection body of a ball, we give an explicit counter-example for the Shephard problem of convex bodies in Rn (n ≥ 3) and an affirmative answer to the question of Zhang.