On the homothety conjecture

Elisabeth Werner, Deping Ye

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

Let K be a convex body in Rn and δ > 0. The homothety conjecture asks: Does Kδ = cK imply that K is an ellipsoid? Here Kδ is the (convex) floating body and c is a constant depending on δ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies Bnp , 1 ≤ p ≤ ∞, the unit balls of ℓnp; namely, we show that (Bn p)δ = cBnp if and only if p = 2. We also show that the homothety conjecture is true for a general convex body K if δ is small enough. This improves earlier results by Schütt and Werner [16] and Stancu [20].

Original languageEnglish (US)
Pages (from-to)1-20
Number of pages20
JournalIndiana University Mathematics Journal
Volume60
Issue number1
DOIs
StatePublished - 2011

Keywords

  • Convex floating body
  • Homothety problem

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