## Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 171-189 |

Number of pages | 19 |

Journal | Beitrage zur Algebra und Geometrie |

Volume | 52 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2011 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology

## Keywords

- Convex set
- Distortion
- Measure of symmetry