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arXiv · 2311.03070

Floating bodies and duality in spaces of constant curvature

Florian Besau·Elisabeth M. Werner
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Abstract

We investigate a natural analog to Lutwak's $p$-affine surface area in $d$-dimensional spherical, hyperbolic and de Sitter space. In particular, we show that these curvature measures appear naturally as the volume derivative of floating bodies of non-Euclidean convex bodies conjugated by duality, such as spherical, hyperbolic and de Sitter convex bodies. We provide a unifying framework by establishing a real-analytic version of this relation controlled by the constant curvature of the $d$-dimensional real space form. These new curvature measures relate in two distinctly different ways to curvature measures on Euclidean space, one of which is Lutwak's centro-affine invariant $p$-affine surface area, and the other is related to a rigid-motion invariant curvature measure that appears naturally as the volume derivative of Schneider's mean-width separation body.

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Florian Besau, Elisabeth M. Werner. 2023-11-06. Floating bodies and duality in spaces of constant curvature. https://arxiv.org/abs/2311.03070

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