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Balancing vectors and convex bodies

Wojciech Banaszczyk (1993)

Studia Mathematica

Let U, V be two symmetric convex bodies in n and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors u 1 , . . . , u n U such that, for each choice of signs ε 1 , . . . , ε n = ± 1 , one has ε 1 u 1 + . . . + ε n u n r V where r = ( 2 π e 2 ) - 1 / 2 n 1 / 2 ( | U | / | V | ) 1 / n . Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence ( u n ) such that the series n = 1 ε n u π ( n ) is divergent for any choice of signs ε n = ± 1 and any permutation π of indices.

Borsuk-Ulam type theorems

Adam Idzik (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

A generalization of the theorem of Bajmóczy and Bárány which in turn is a common generalization of Borsuk's and Radon's theorem is presented. A related conjecture is formulated.

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