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Ambiguous loci of the farthest distance mapping from compact convex sets

100%

De Blasi F. S. , Myjak J.

tom 112

nr 2

99-107

Let 𝔼 be a strictly convex separable Banach space of dimension at least 2. Let K(𝔼) be the space of all nonempty compact convex subsets of 𝔼 endowed with the Hausdorff distance. Denote by $K^0$ the set of all X ∈ K(𝔼) such that the farthest distance mapping $a ↦ M_X(a)$ is multivalued on a dense subset of 𝔼. It is proved that $K^0$ is a residual dense subset of K(𝔼).

Upper Estimate of Concentration and Thin Dimensions of Measures

80%

Gacki H. , Lasota A. , Myjak J.

nr 2

149-162

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.