Porosity and unique completion in strictly convex spaces
The author studies the family \(G\) of all closed convex sets with a unique extension to a complete set: a subset \(S\) of a Minkowski space with a strictly convex norm is an element of \(G\) if and only if \(S\) admits a unique maximal superset of the same diameter. This family is extensively studied starting from [\textit{V. G.~Boltyanskii}, Colloq. Math. 21, 253--263 (1970; Zbl 0198.27102)], and the author contributes two facts about \(G\), opposite in their spirit. The first one improves the result of \textit{H.~Groemer} [Geom. Dedicata 20, 319--334 (1986; Zbl 0587.52002)]: \(G\) is not only nowhere dense in the Hausdorff metric, but is also lower porous, i.e. there exists a constant \(C>0\), such that every ball of large radius \(R\) contains a ball of radius \(R/C\) outside \(G\). The second result is a stochastic construction procedure which provides a maximal element of \(G\) (i.e. a complete set) with probability one, see [\textit{F.~Bavaud}, Trans. Am. Math. Soc. 333, No. 1, 315--324 (1992; Zbl 0760.52004)] for the special case of the Euclidean plane.
scientific article; zbMATH DE number 5859311
scientific article