Vol. 52, No. 2, 1974

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Some content maximizing properties of the regular simplex

R. Michael Tanner

Vol. 52 (1974), No. 2, 611–616

In this paper it is shown that the regular simplex maximizes the sum of the squared contents of all i-dimensional faces, for all i = 2, , n, when the sum of the one-dimensional squared contents is fixed. An immediate corollary is that the regular simplex has the largest total length of all joining lines, total area of all triangles, total volume of all tetrahedra, and so forth, for a fixed sum of squared line lengths. Some related unsolved conjectures are presented.

Mathematical Subject Classification 2000
Primary: 52A40
Received: 8 August 1973
Published: 1 June 1974
R. Michael Tanner