Floating and Illumination Bodies for Polytopes: Duality Results

Duality of Floating and Illumination Bodies

We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question for the class of centrally symmetric convex bodies. We provide precise estimates for $(B_p^n)$ and for centrally symmetric convex bodies with everywhere positive Gauss curvature. Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids.

Floating body, illumination body, and polytopal approximation

Let $K$ be a convex body in $\Bbb R^{d}$ and $K_{t}$ its floating bodies. There is a polytope with at most $n$ vertices that satisfies $$ K_{t} \subset P_{n} \subset K $$ where $$ n \leq e^{16d} \frac{vol_{d}(K \setminus K_{t})}{t\ vol_{d}(B_{2}^{d})} $$ Let $K^{t}$ be the illumination bodies of $K$ and $Q_{n}$ a polytope that contains $K$ and has at most $n$ $d-1$-dimensional faces. Then $$ vol_{d}(K^{t} \setminus K) \leq cd^{4} vol_{d}(Q_{n} \setminus K) $$ where $$ n \leq \frac{c}{dt} \ vol_{d}(K^{t} \setminus K) $$

Flag numbers and floating bodies

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.

马勒猜想与浮体之间的联系

A connection between the Mahler conjecture and floating bodies

本文证明了浮体不光滑的多面体结构结果，并将其应用于体积乘积极小值组合结构的研究，建立了马勒猜想与浮体的联系。

Random polytopes: central limit theorems for intrinsic volumes

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein's method from probability theory.

Separation bodies: a conceptual dual to floating bodies

Let $K$ be a convex body in Euclidean space ${\mathbb R}^d$, and let a translation invariant, locally finite Borel measure on the space of hyperplanes in ${\mathbb R}^d$ be given. For $δ\ge 0$, we consider the set of all points $x$ for which the set of hyperplanes separating $K$ and $x$ has measure at most $δ$. This defines the separation body of $K$, with respect to the given measure and the parameter $δ$. Separation bodies are meant as conceptual duals to floating bodies, and they are expected to play a role in the investigation of random polytopes generated as intersections of random halfsp…