Original Paper

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry

, Volume 52, Issue 1, pp 171-189

First online:

Fine structure of convex sets from asymmetric viewpoint

  • Gabor TothAffiliated withDepartment of Mathematics, Rutgers University Email author 

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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.

Keywords

Convex set Distortion Measure of symmetry

Mathematics Subject Classification (2000)

52A05 52A38 52B11