Floating bodies and duality in spaces of constant curvature

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 du...

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Bibliographic Details
Published in:arXiv.org 2023-11
Main Authors: Besau, Florian, Werner, Elisabeth M
Format: Article
Language:English
Subjects:
Curvature
Euclidean geometry
Floating bodies
Invariants
Surface area
Online Access:Get full text
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Summary: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.
ISSN:2331-8422