### A characterisation of the ellipsoid in terms of concurrent sections.

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Let K be a convex body in ${\mathbb{R}}^{n}$ and B be the Euclidean unit ball in ${\mathbb{R}}^{n}$. We show that $li{m}_{t\to 0}\left(\right|K|-|{K}_{t}\left|\right)/\left(\right|B|-|{B}_{t}\left|\right)=as\left(K\right)/as\left(B\right)$, where as(K) respectively as(B) is the affine surface area of K respectively B and ${{K}_{t}}_{t\ge 0}$, ${{B}_{t}}_{t\ge 0}$ are general families of convex bodies constructed from K,B satisfying certain conditions. As a corollary we get results obtained in [M-W], [Schm], [S-W] and [W].

We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in ${\mathbb{R}}^{m\times n}$, $min(m,n)\le 2$, then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this...