Minkowski Measure of Asymmetry and Minkowski Distance for Convex Bodies

University dissertation from Uppsala : Matematiska institutionen

Abstract: This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let C and D be n-dimensional convex bodies. Denote by As(C) and As(D) the Minkowski measures of asymmetry of C and D resp. and by d(C,D) the Minkowski distance between C and D.In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that d(C,D) < n(As(C) + As(D))/2 for all convex bodies C,D.In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, d(C,D) < (n-1) min(As(C),As(D)) + n.In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form conv(C,p) where C is a convex set with known measure of asymmetry and p is a point outside C.In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) > n - ? for 0 < ? < 1/? where ? = 8(n+1), then d(C,S) < 1 + 8(n+1) ? .