October 1, 2025Open Access

Floating bodies for ball-convex bodies

Key Points

The study defines floating bodies within the framework of ball-convex bodies, expanding geometric understanding.
It introduces relative affine surface area as a measure derived from the volume of floating bodies, highlighting its geometric significance.
This relative affine surface area is shown to be upper semi continuous and invariant under rigid motions, indicating its robustness.
Geometric properties of these bodies provide essential insights into the relationship between volume and surface area in higher dimensions.

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.

Floating bodies for ball-convex bodies

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Abstract

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