Floating body, illumination body, and polytopal approximation

Abstract

Let $K$ be a convex body in ${\mathbb{R}}^d$ 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 $$n\le e^{16d}\frac{vo{l}_d(K\setminus K_t)}{tvo{l}_d(B_2^d)}$$ 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 $$vo{l}_d(K^t\setminus K)\le c{d}^{4}vo{l}_d(Q_n\setminus K)$$ where $$n\le \frac{c}{dt}vo{l}_d(K^t\setminus K)$$

Cited by in corpus (2)

• Data depth and floating body
• Illumination depth