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On Bárány's theorems of Carathéodory and Helly type

Ehrhard Behrends (2000)

Studia Mathematica

The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows:...

On order and metric convexities in $Z^{n}$

M. Changat, A. Vijayakumar (1992)

Compositio Mathematica

On point sets fixing a convex body from within.

Braß, Peter, Herburt, Irmina (1997)

Beiträge zur Algebra und Geometrie

On starshapedness of the union of closed sets in $R^{n}$

Krzysztof Kołodziejczyk (1987)

Colloquium Mathematicae

On the estension of Lipschitz functions with respect to two Hilbert norms and two Lipschitz conditions

Chandan S. Vora (1973)

Rendiconti del Seminario Matematico della Università di Padova

Optimal bounds for the colored Tverberg problem

Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler (2015)

Journal of the European Mathematical Society

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.