Fine structure of convex sets from asymmetric viewpoint

We study a sequence of measures of symmetry {σ_m(L, O)}m≥1 for a convex body L with a specified interior point O in an n-dimensional Euclidean vector space ε. The mth term σ_m(L, O) measures how far the m-dimensional affine slices of L (across O) are from an m-simplex (viewed from O). The interior of L naturally splits into regular and singular sets, where the singular set consists of points O with largest possible σ_n(L, O). In general, to calculate the singular set is difficult. In this paper we derive a number of results that facilitate this calculation. We show that concavity of σ_n(L, .) viewed as a function of the interior of L occurs at points O with highest degree of singularity, or equivalently, at points where the sequence {σ_m(L, O)}m≥1 is arithmetic. As a byproduct, these results also shed light on the structure and connectivity properties of the regular and singular sets.