Centroid bodies and the convexity of area functionals

Abstract

Abstract. We introduce a new volume definition on normed vector spaces. We show that the induced k-area functionals are convex for all k. In the particular case k = 2, our theorem implies that Busemann's 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is related to the centroid body and prove some affine isoperimetric inequalities.

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

“…We will follow the approach of [LW17b] based on Jacobians. The reader is referred to [Iva09, APT04,Ber14] for other equivalent viewpoints.…”

Intrinsic structure of minimal discs in metric spaces

“…While there is an essentially unique natural way to measure areas of Riemannian surfaces, there are many different ways to measure areas of Finsler surfaces, some of them more appropriate for different questions. We refer the reader to [Iva08], [Ber14], [LW16c], [APT04] and the literature therein for more information.…”

Energy and area minimizers in metric spaces

“…Remark 2.1. We refer to another similar geometric interpretation of a definition of area discussed in [Ber14].…”

“…The most prominent examples are the Busemann (or Hausdorff) definition µ b , the Holmes-Thompson definition µ ht , the Benson (or Gromov mass * ) definition m * and the inscribed Riemannian (or Ivanov) definition µ i . We refer to [APT04] for a thorough discussion of these examples and of the whole subject; and to [Iva08], [BI12], [Ber14] for recent developments. Here, we just mention the Jacobians of these four examples (cf.…”