The Floating Body Problem

Abstract

Let K be a convex body with boundary of class C 4 . We prove that, for a sufficiently small δ, the floating body K δ is homothetic to K if and only if K is an ellipsoid. The proof relies on properties of the affine curvature flow.

Halfspace depth and floating body

“…, with equality if and only if K is an ellipsoid (see, e.g., [156,Section 10.5]). The affine isoperimetric inequality is stronger than the classical isoperimetric inequality and provides solutions to many problems where ellipsoids are extrema [106,163,171,183]. The affine isoperimetric inequality (18) is equivalent to another classical inequality from convex geometry, the Blaschke-Santaló inequality [17,153].…”

Convex floating bodies as approximations of Bergman sublevel sets on tube domains

“…Inspired by a construction due to Dupin, these were first introduced by Schütt and Werner (in [12]) as a tool for extending the notion of Blaschke's surface area measure to nonsmooth convex boundaries. Since its introduction, the floating body has made appearances in the context of polyhedral approximations (see [11]), the homethety conjecture (see [14] and [15]) and, more recently, the hyperplane conjecture (in [3]). Now, let Ω := {x + iy ∈ C n : y ∈ D}.…”

On the Stability of the p-Affine Isoperimetric Inequality

“…There are several important contributions of geometric flows to convex geometry, for example, a proof of the affine isoperimetric inequality by B. Andrews using the affine normal flow [1], obtaining the necessary and sufficient conditions for the existence of a solution to the discrete L 0 -Minkowski problem using crystalline curvature flow by A. Stancu [35,36,39] and independently by B. Andrews [3], and a proof of the p-affine isoperimetric inequality in the class of origin-symmetric convex bodies in R 2 using the affine normal [19]. See [37,38,40,41] for more applications of flows, in particular, a newly defined family of centro-affine p-flows and their applications to centro-affine differential geometry by A. Stancu [40,41].…”