Convex Bodies In Euclidean Space Research Articles

Cyclic inequalities for generalized mixed width-integrals

In this paper, we research the cyclic inequalities of generalized width-integral in convex geometry analysis. Based on previous studies, we use Hölder’s inequality and inverse Hölder’s inequality to study further the generalized width-integral inequalities of the convex body in Euclidean space, establish a series of cyclic inequalities, and give the condition of equality sign.

Separation bodies: a conceptual dual to floating bodies

Let K be a convex body in Euclidean space $${\mathbb R}^d$$ , and let a translation invariant, locally finite Borel measure on the space of hyperplanes in $${{\mathbb {R}}}^d$$ be given. For $$\delta \ge 0$$ , we consider the set of all points x for which the set of hyperplanes separating K and x has measure at most $$\delta $$ . This defines the separation body of K, with respect to the given measure and the parameter $$\delta $$ . Separation bodies are meant as conceptual duals to floating bodies, and they are expected to play a role in the investigation of random polytopes generated as intersections of random halfspaces, in a similar way that floating bodies are useful for studying convex hulls of random points. After discussing some elementary properties of separation bodies, we carry out first examples to this effect.

ON THE MAGNITUDE AND INTRINSIC VOLUMES OF A CONVEX BODY IN EUCLIDEAN SPACE

Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is closely related to intrinsic volumes. Here we prove an upper bound for the magnitude of a convex body in Euclidean space in terms of its intrinsic volumes. The result is deduced from an analogous known result for magnitude in $\ell_1^N$, via approximate embeddings of Euclidean space into high-dimensional $\ell_1^N$ spaces. As a consequence, we deduce a sufficient condition for infinite-dimensional subsets of a Hilbert space to have finite magnitude. The upper bound is also shown to be sharp to first order for an odd-dimensional Euclidean ball shrinking to a point; this complements recent work investigating the asymptotics of magnitude for large dilatations of sets in Euclidean space.

Lower and upper bounds for the waists of different spaces

We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space. We establish waist-type results in terms of the Hausdorff measure.

Flag numbers and floating bodies

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure.Our results are applied in spherical and hyperbolic space. This leads to new asymptotic results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.

Relative convex body in Euclidean and hyperbolic spaces

This paper is to generalize the concept of convex body to the so called relative convexbody in Euclidean space 𝐸􀯡 . Some geometrical and topological properties for this kind of sets arediscussed. Some properties of the central projection map(Beltrami map) introduced to discuss theseconcepts in the hyperbolic space 𝐻􀯡 ..

A Fourier analytic proof of the Blaschke-Santaló Inequality

The Blaschke-Santaló Inequality is the assertion that the volume product of a centrally symmetric convex body in Euclidean space is maximized by (and only by) ellipsoids. In this paper we give a Fourier analytic proof of this fact.

Upper bounds for packings of spheres of several radii

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

Local Tensor Valuations

The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant Minkowski tensors that were introduced by McMullen and characterized by Alesker. In analogy to the characterization theorems of Hadwiger and Alesker, we give here a complete classification of all locally defined tensor measures on convex bodies that share with the local Minkowski tensors the basic geometric properties of isometry covariance and weak continuity.

Reduced convex bodies in Euclidean space—A survey

A convex body R in Euclidean space E d is called reduced if the minimal width Δ ( K ) of each convex body K ⊂ R different from R is smaller than Δ ( R ) . This definition yields a class of convex bodies which contains the class of complete sets, i.e., the family of bodies of constant width. Other obvious examples in E 2 are regular odd-gons. We know a relatively large amount on reduced convex bodies in E 2 . Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For d ≥ 3 we do not even have tools which permit us to recognize what the boundary of R looks like. The class of reduced convex bodies has interesting applications. We present the current state of knowledge about reduced convex bodies in E d , recall some striking related research problems, and put a few new questions.

Verification of polytopes by brightness functions

We show that in the class of origin-centered convex bodies in Euclidean space of dimension at least three, a polytope is uniquely determined by its brigthness function in a suitably chosen, but very small set of directions.

Rotation Invariant Minkowski Classes of Convex Bodies

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝ n which is closed under Minkowski addition and non-negative dilatations. A convex body in ℝ n is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T 1 , T 2 such that M + T 1 = T 2 , and T 1 , T 2 belong to the rotation invariant Minkowski class generated by K . It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K , which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

Affine diameters of convex bodies—a survey

This survey covers various topics related to affine diameters of convex bodies in Euclidean space. It describes the known properties of affine diameters and contains a list of existing and new open problems. Also, characterizations of simplices and centrally symmetric convex bodies, as well as a description of a measure of symmetry in terms of affine diameters, are given.

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann [3] and Hug [6], who use much less elementary techniques than the methods of this paper. In fact, Hann [4] asked for a simpler proof of these results ([4], Problem 2 on p. 900). We remark that our formula can be considered as a Euclidean analogue of a spherical result proved in [2], p. 46, and that our method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensional Minkowski spaces, see Hann [5], p. 363. For further information about the notions we use below we refer to Schneider’s book [9]. Let K be the set of all convex bodies in Euclidean space R, i.e. the set of all compact, convex, non-empty subsets of R. Let Sn−1 be the unit sphere. For K ∈ K, let Nor K be the set of all support elements ofK, i.e. the pairs (x, u) ∈ Rn×Sn−1 such that x is a boundary point of K and u is an outer unit normal vector of K at the point x. The support measures (or generalized curvature measures) of K, denoted by Θ0(K, ·), . . . ,Θn−1(K, ·), are the unique Borel measures on R × Sn−1 that are concentrated on Nor K and satisfy ∫

Characterizations via linear combinations of curvature measures

A convex body in euclidean space with one curvature measure being a fixed multiple of the boundary measure is a ball. This result of R. Schneider is generalized to linear combinations of curvature measures with positive coefficients. The proof relies on generalized Minkowski integral formulas, a volume representation via the interior reach function, and the isoperimetric inequality.

On the mean value of the area of a random polygon in a plane convex body

Let K be a convex body in Euclidean space R d , d ≥2, with volume V ( K ) = 1, and n ≥ d +1 be a natural number. We select n independent random points y 1 , y 2 , …, y n from K (we assume they all have the uniform distribution in K ). Their convex hull co { y 1 , y 2 , …, y n } is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope It is easy to see that if U : R d → R d is a volume preserving affine transformation, then for every convex body K with V ( K ) = 1, m ( K, n ) = m ( U ( K ), n ).

Finite dimensional polish spaces are extreme boundaries of convex bodies in Euclidean space

It is proved that everyn-dimensional Polish space is homeomorphic to the set of extreme points of a compact convex set inR18(n+1).

Isometries of the Space of Convex Bodies in Euclidean Space

Isometries of the Space of Convex Bodies in Euclidean Space

The directions of the line segments and of the r ‐dimensional balls on the boundary of a convex body in Euclidean space

The directions of the line segments and of the <i>r</i> ‐dimensional balls on the boundary of a convex body in Euclidean space