Ulam Floating Body

TL;DR

This paper introduces a new convex body construction related to floating bodies, explores its properties and connections to $p$-affine surface areas, and discusses its implications for Ulam's floating body problem.

Contribution

It presents a novel construction of bodies isomorphic to floating bodies, analyzes their properties, and links them to longstanding questions in convex geometry.

Findings

New construction of bodies related to floating bodies

Connections established with $p$-affine surface areas

Implications for Ulam's floating body problem

Abstract

We study a new construction of bodies from a given convex body in $\mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. We show that these bodies are related to Ulam's long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.