Date: 30 Mar 2011

Fine structure of convex sets from asymmetric viewpoint

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Abstract

We study a sequence of measures of symmetry \({\{\sigma_m(\mathcal {L}, \mathcal {O})\}_{m\geq 1}}\) for a convex body \({\mathcal {L}}\) with a specified interior point \({\mathcal {O}}\) in an n-dimensional Euclidean vector space \({\mathcal {E}}\) . The mth term \({\sigma_m(\mathcal {L}, \mathcal {O})}\) measures how far the m-dimensional affine slices of \({\mathcal {L}}\) (across \({\mathcal {O}}\) ) are from an m-simplex (viewed from \({\mathcal {O}}\) ). The interior of \({\mathcal {L}s}\) naturally splits into regular and singular sets, where the singular set consists of points \({\mathcal {O}}\) with largest possible \({\sigma_n(\mathcal {L}, \mathcal{O})}\) . 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 \({\sigma_n(\mathcal {L},.)}\) viewed as a function of the interior of \({\mathcal {L}}\) occurs at points \({\mathcal {O}}\) with highest degree of singularity, or equivalently, at points where the sequence \({\{\sigma_m(\mathcal {L}, \mathcal {O})\}_{m\geq 1}}\) is arithmetic. As a byproduct, these results also shed light on the structure and connectivity properties of the regular and singular sets.