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NORMAL TILINGS OF A BANACH SPACE AND ITS BALL
Author(s) -
Deville Robert,
GarcíaBravo Miguel
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12043
Subject(s) - mathematics , unit sphere , banach space , regular polygon , disjoint sets , uniformly convex space , bounded function , combinatorics , convex body , ball (mathematics) , reflexive space , separable space , pure mathematics , banach manifold , mathematical analysis , interpolation space , lp space , convex hull , geometry , functional analysis , biochemistry , chemistry , gene
We show some new results about tilings in Banach spaces. A tiling of a Banach space X is a covering by closed sets with non‐empty interior, so that they have pairwise disjoint interiors. If, moreover, the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal. In 2010, Preiss constructed a convex normal tiling of the separable Hilbert space. For Banach spaces with Schauder basis, we will show that Preiss' result is still true with starshaped tiles instead of convex ones. Also, whenever X is uniformly convex we give precise constructions of convex normal tilings of the unit sphere, the unit ball or in general of any convex body.