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On a Property of $n$-Dimensional Simplices

M. V. Nevskij

P. G. Demidov Yaroslavl State University

Abstract: Suppose that $n\in\mathbb N$ and $S$ is a simplex in $\mathbb R^n$, containing the cube $[0,1]^n$. It is proved that, for some $i=1,\dots,n$, the simplex $S$ contains an interval of length $n$ parallel to the $i$th coordinate axis. The relationship with questions arising in linear interpolation of continuous functions of $n$ variables is noted.

Keywords: $n$-dimensional simplex, polytope, barycentric coordinates, axial diameter, interpolation projection operator, Steiner symmetrization, Hadamard number.

UDC: 517.51+514.17

Received: 22.04.2009
Revised: 03.07.2009

DOI: 10.4213/mzm7698

 English version: Mathematical Notes, 2010, 87:4, 543–555

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