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Farthest Points and Strong Convexity of Sets
G. E. Ivanov Moscow Institute of Physics and Technology
Abstract:
We consider the existence and uniqueness of the farthest point of a given set
$A$ in Banach space
$E$ from a given point
$x$ in the space
$E$. It is assumed that
$A$ is a convex, closed, and bounded set in a uniformly convex Banach space
$E$ with Fréchet differentiable norm. It is shown that, for any point
$x$ sufficiently far from the set
$A$, the point of the set
$A$ which is farthest from
$x$ exists, is unique, and depends continuously on the point
$x$ if and only if the set
$A$ in the Minkowski sum with some other set yields a ball. Moreover, the farthest (from
$x$) point of the set
$A$ also depends continuously on the set
$A$ in the sense of the Hausdorff metric. If the norm ball of the space
$E$ is a generating set, these conditions on the set
$A$ are equivalent to its strong convexity.
Keywords:
optimization problem, farthest points, strong convexity of a set, Banach space, Fréchet differentiable norm, Minkowski sum, Hausdorff metric, metric antiprojection, antisun.
UDC:
517.982.252+
517.982.256 Received: 05.01.2008
Revised: 15.08.2009
DOI:
10.4213/mzm4428
Fulltext:
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