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arXiv · 1305.1437

Centroid bodies and the convexity of area functionals

Andreas Bernig
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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.

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Andreas Bernig. 2013-05-07. Centroid bodies and the convexity of area functionals. https://arxiv.org/abs/1305.1437

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