IUMJ

Title: On $L_p$-affine surface areas

Authors: Elisabeth Werner

Issue: Volume 56 (2007), Issue 5, 2305-2324

Abstract:

Let $K$ be a convex body in $\mathbb{R}^n$ with centroid at $0$ and $B$ be the Euclidean unit ball in $\mathbb{R}^n$ centered at $0$. We show that \[ \lim_{t \to 0} \frac{|K| - |K_t|}{|B| - |B_t|} = \frac{O_p(K)}{O_p(B)}, \] where $|K|$ respectively $|B|$ denotes the volume of $K$ respectively $B$, $O_p(K)$ respectively $O_p(B)$ is the $p$-affine surface area of $K$ respectively $B$ and $\{ K_t \}_{t \geq 0}$, $\{ B_t \}_{t \geq 0}$ are general families of convex bodies constructed from $K$, $B$ satisfying certain conditions. As a corollary we get results obtained in [M. Meyer and E. Werner, \textit{On the $p$-affine surface area}, Adv. Math. \textbf{152} (2000), Number 2, 288--313; C. Sch\"utt and E. Werner, \textit{Polytopes with vertices chosen randomly from the boundary of a convex body}, in: "Geometric Aspects of Functional Analysis," Lecture Notes in Math. \textbf{1807} (Berlin: Springer, 2003), 241--422; C. Sch\"utt and E. Werner, \textit{Surface bodies and $p$-affine surface area}, Adv. Math. \textbf{187} (2004), Number 1, 98--145; E. Werner, \textit{The $p$-affine surface area and geometric interpretations}, in: "IV International Conference in Stochastic Geometry, Convex Bodies, Empirical Measures and Applications to Engineering Science, Vol. II (Tropea, 2001)," Rend. Circ. Mat. Palermo (2) Suppl., Number 70 (2002), 367--382].