Abstract
Abstract We introduce a measure of (central) symmetry for a convex body K in ℝn based on the volume of the harmonic mean of K and −K (that is, the body whose gauge is the average of the gauges of K and −K), and compare it to other classical volumetric measures of symmetry. We prove sharp inequalities between the volume of the harmonic mean of K and −K and the volume of their intersection, and we show that, if K is in John’s position, then the harmonic mean of K and −K has the smallest volume exactly when K is the Euclidean ball and the largest volume exactly when K is a simplex.
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