Vol. 31, No. 3, 1969

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The content of some extreme simplexes

David Slepian

Vol. 31 (1969), No. 3, 795–808
Abstract

Formulae are presented that give the content of a simplex in Euclidean n-space: (i) in terms of the lengths of and the angles between the vectors from a fixed point to the vertices of the simplex; (ii) in terms of the lengths of and the angles between the perpendiculars from a fixed point to the bounding faces of the simplex. We then determine the largest simplex whose vertices are given distances from a fixed point and we determine the smallest simplex whose faces are given distances from a fixed point. As special cases we find that the regular simplex is the largest simplex contained in a given sphere and is also the smallest simplex containing a given sphere.

Mathematical Subject Classification
Primary: 52.40
Milestones
Received: 14 February 1969
Published: 1 December 1969
Authors
David Slepian