Gaussian behaviour and average of marginals for convex bodies.
In this paper we give general methods of construction of various equivalent minimal pairs of compact convex sets that are not translates of one another.
Let ε > 0 and 1 ≤ k ≤ n and let be affine subspaces of ℝⁿ, each of dimension at most k. Let if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map such that for all 1 ≤ l ≤ p and we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y.
Using Baire categories uniqueness of geodesic segments and existence of closed geodesics on typical convex surfaces are investigated.
We solve problems of Buffon type for a lattice with elementary tile a nonconvex polygon, using as test bodies a line sigment and a circle.