Remarks on minkowski symmetrizations

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

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).

Estimates for measures of lower dimensional sections of convex bodies

The geometry of smooth convex hypersurfaces in euclidean n-space Rn on the one hand and the geometry of the boundary of arbitrary convex bodies in Rn on the other are closely related (see [1] §17). The former belongs to differential geometry, the latter to geometric convexity. Some theorems have a differential geometric version as well as a convexity version; these depend on each other.

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.

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).

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.

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.

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.

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

Hyperspaces of smooth convex bodies up to congruence

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

A generalized localization theorem and geometric inequalities for convex bodies

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 .