Sections of convex bodies in Johns and minimal surface area position

We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in Johns position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in Johns position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.

Random approximation and the vertex index of convex bodies

We prove that there exists an absolute constant $alpha >1$ with the following property: if $K$ is a convex body in ${mathbb R}^n$ whose center of mass is at the origin, then a random subset $Xsubset K$ of cardinality ${rm card}(X)=lceilalpha nrceil $ satisfies with probability greater than $1-e^{-n}$ {Ksubseteq c_1n,{mathrm conv}(X),} where $c_1>0$ is an absolute constant. As an application we show that the vertex index of any convex body $K$ in ${mathbb R}^n$ is bounded by $c_2n^2$, where $c_2>0$ is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.