Floating bodies for ball-convex bodies

TL;DR

This work extends floating bodies and affine surface area to the setting of ball-convex geometry by introducing the $R$-ball floating body $K_\delta^R$ for $R$-ball convex bodies and defining the relative affine surface area $\mathrm{as}^R(K)$ as the right derivative of the volume deficit. The authors establish a precise limit: $\lim_{\delta\to0} (\mathrm{vol}_n(K)-\mathrm{vol}_n(K_\delta^R))/\delta^{2/(n+1)}$ equals a curvature-based integral over the boundary, providing a rigid-motion invariant, upper semi-continuous valuation $\mathrm{as}^R(K)$ which generalizes classical affine surface area (recovered as $R\to\infty$). They also extend the construction to $L$-convex bodies via $\mathrm{as}^L(K)$ and provide an integral identity used in the evaluation of constants, along with properties such as homogeneity, a valuation formula, and affine-isoperimetric-type bounds. The results pave the way for further $L_p$-extensions and potential applications in approximation theory for ball-convex bodies.

Abstract

We define floating bodies in the class of $n$-dimensional ball-convex bodies. A right derivative of volume of these floating bodies leads to a surface area measure for ball-convex bodies which we call relative affine surface area. We show that this quantity is a rigid motion invariant, upper semi continuous valuation.

Figures (1)

• Figure 1: The angles

Theorems & Definitions (14)

• Definition 1
• Theorem 1
• Definition 2
• Proposition 2
• Proposition 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof