Mean Minkowski Measures

Abstract

In this final chapter we introduce a sequence of new measures of symmetry $\{\boldsymbol{\sigma }_{k}\}_{k\geq 1}$. For a convex body $\mathcal{C}$, the kth term $\boldsymbol{\sigma }_{k} =\boldsymbol{\sigma } _{\mathcal{C},k}$, k ≥ 1, is a function on the interior of $\mathcal{C}$. For an interior point O of $\mathcal{C}$, $\boldsymbol{\sigma }_{k}(O)$ measures how far are the k-dimensional affine slices of $\mathcal{C}$ (across O) from a k-simplex (viewed from O). The minimum value of $\boldsymbol{\sigma }_{k}(O)$ is 1 corresponding to a k-dimensional simplicial slice of $\mathcal{C}$. For k ≥ 2, the maximum value of $\boldsymbol{\sigma }_{k}(O)$ corresponds to symmetric $\mathcal{C}$ with respect to O. In this section we derive a host of arithmetic properties of the sequence $\{\boldsymbol{\sigma }_{k}\}_{k\geq 1}$, and show various connections with the maximal distortion $\mathfrak{m}_{\mathcal{C}}$, and the Minkowski measure $\mathfrak{m}_{\mathcal{C}}^{{\ast}}$.