In this note we show that a non-degenerated polytope in IRn with n+k, 1 ≤ k < n, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes. 0 Introduction and notations The canonical Euclidean inner product in IR is denoted by 〈·, ·〉, the norm in `p is denoted by ‖ · ‖p, 1 ≤ p ≤ ∞. By a convex body K ⊂ IR we shall always ...

Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel [Mathematika 52 (2005), 47–52] introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce a relative of the covering parameter called covering index, which turn...

Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bo...

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The main result of this paper is a proof of the Illumination Conjecture for “fat” spindle convex b...

The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Here we show corresponding inequalities for arbitrary convex bodies, where the successive minima are replaced by certain successive diameters and successive widths. We further give some applications of these results to successive radii, intrinsic volumes and the latt...

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points o...

The covering number c(K) of a convex body K is the least number of smaller homothetic copies of K needed to cover K . We provide new upper bounds for c(K) when K is centrally symmetric by introducing and studying the generalized α -blocking number βα 2 (K) of K . It is shown that when a centrally symmetric convex body K is sufficiently close to a centrally symmetric convex body K′ , then c(K) i...

A circle C holds a convex body K ⊂ R3 if K can’t be moved far away from its position without intersecting C . One of our results says that there is a convex body K ⊂ R3 such that the set of radii of all circles holding K has infinitely many components. Another result says that the circle is unique in the sense that every frame different from the circle holds a convex body K (actually a tetrahed...

We investigate the effect of a Steiner type symmetrization on the isotropic constant of a convex body. We reduce the problem of bounding the isotropic constant of an arbitrary convex body, to the problem of bounding the isotropic constant of a finite volume ratio body. We also add two observations concerning the slicing problem. The first is the equivalence of the problem to a reverse Brunn-Min...

We deal with different properties of a smooth and strictly convex body that depend on the behavior of the planar sections of the body parallel to and close to a given tangent plane. The first topic is boundary points where any given convex domain in the tangent plane can be approximated by a sequence of suitably rescaled planar sections (so-called p-universal points). In the second topic, the g...