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A general geometric construction for affine surface area

Elisabeth Werner (1999)

Studia Mathematica

Let K be a convex body in n and B be the Euclidean unit ball in n . We show that l i m t 0 ( | K | - | K t | ) / ( | B | - | B t | ) = a s ( K ) / a s ( B ) , where as(K) respectively as(B) is the affine surface area of K respectively B and K t t 0 , B t t 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].

A note on equality of functional envelopes

Martin Kružík (2003)

Mathematica Bohemica

We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in m × n , min ( m , n ) 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.

A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets

Gilbert Crombez (2006)

Czechoslovak Mathematical Journal

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...

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