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We prove that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies K t , is a convex function of t. Thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain… (More)

There are two positive, absolute constants c 1 and c 2 so that the volume of the difference set of the d-dimensional Euclidean ball and an inscribed polytope with n vertices is larger than c 2 d n − 2 d−1 vol d (B d 2) for n ≥ (c 1 d) d−1 2 .

It is proved that for a symmetric convex body K in R n , if for some τ > 0, |K ∩ (x + τ K)| depends on x K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies are studied.

The problem of the existence of an equi-partition of a curve in R n has recently been raised in the context of computational geometry (see [2] and [3]). The problem is to show that for a (continuous) curve Γ : [0, 1] → R n and for any positive integer N, there exist points t 0 = 0 < t 1 <. .. < t N −1 < 1 = t N , such that d(Γ(t i−1), Γ(t i)) = d(Γ(t i),… (More)

Let P be a convex polytope in R d , d = 3 or 2, with n vertices. We present linear time algorithms for approximating P by simpler polytopes. For instance, one such algorithm selects k < n vertices of P whose convex hull is the approximating polytope. The rate of approximation, in the Hausdorff distance sense, is best possible in the worst case. An analogous… (More)