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Porosity and unique completion in strictly convex spaces

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Abstract

We are concerned in this paper with topological and stochastic properties of the family \({\mathcal G}\) of all closed convex sets with a unique extension to a complete set. With the help of a strengthened version of a lemma by Groemer we show that, in Minkowski spaces with a strictly convex norm, \({\mathcal G}\) is lower porous. This improves a previous result from Groemer (Geom. Dedicata 20:319–334, 1986) where, in the same context, \({\mathcal G}\) was proved to be nowhere dense. In contrast to this fact we show that, in these spaces, there is a stochastic construction procedure which provides a complete set with probability one. This generalizes an earlier result of Bavaud (Trans. Amer. Math. Soc. 333(1):315–324, 1992) proved for the particular case of the Euclidean plane.

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  1. Dpto. Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    J. P. Moreno

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  1. J. P. Moreno
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Correspondence to J. P. Moreno.

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Partially supported by MTM 2009-07848.

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Moreno, J.P. Porosity and unique completion in strictly convex spaces. Math. Z. 267, 173–184 (2011). https://doi.org/10.1007/s00209-009-0615-7

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