### Generic well-posedness in minimization problems.

Ioffe, A., Lucchetti, R.E. (2005)

Abstract and Applied Analysis

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Ioffe, A., Lucchetti, R.E. (2005)

Abstract and Applied Analysis

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de Blasi, F.S., Zhivkov, N.V. (2005)

Abstract and Applied Analysis

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Dušan Holý (2007)

Mathematica Slovaca

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Tadeusz Rzeżuchowski (2012)

Open Mathematics

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We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.

De Blasi, F. (1997)

Serdica Mathematical Journal

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Let E be an infinite dimensional separable space and for e ∈ E and X a nonempty compact convex subset of E, let qX(e) be the metric antiprojection of e on X. Let n ≥ 2 be an arbitrary integer. It is shown that for a typical (in the sence of the Baire category) compact convex set X ⊂ E the metric antiprojection qX(e) has cardinality at least n for every e in a dense subset of E.

Howlett, P.G., Zaslavski, A.J. (2005)

Abstract and Applied Analysis

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Jean-Paul Penot (2003)

Control and Cybernetics

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Marchini, Elsa (2002)

Serdica Mathematical Journal

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We prove that in some classes of optimization problems, like lower semicontinuous functions which are bounded from below, lower semi-continuous or continuous functions which are bounded below by a coercive function and quasi-convex continuous functions with the topology of the uniform convergence, the complement of the set of well-posed problems is σ-porous. These results are obtained as realization of a theorem extending a variational principle of Ioffe-Zaslavski.