Abstract
It is clear that the longest line segment contained in a planar triangle is one of the sides. In this paper we consider the generalization of this statement to Euclideann-space, i.e., we are concerned with the validity of the proposition P(n): A hyperplane section of ann-simplex with maximum volume is a face of the simplex. Eggleston [1] proved that P(3) is valid and we prove here (i) that P(4) is valid, and (ii) a theorem asserting that P(n) false implies P(n + 1) false. However, P(5) is false, as Walkup [3] has shown, and so the question is settled for alln.
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References
H.G. Eggleston, “Plane section of a tetrahedron”,The American Mathematical Monthly 70 (1963) 1108.
H.G. Eggleston, B. Grundaum and V. Klee, “Some semicontinuity theorems for convex polytopes and cell-complexes”,Commentarii Mathematici Helvetici 39 (1964) 165–188.
D.W. Walkup, “A simplex with a large cross section”,The American Mathematical Monthly 75 (1968) 34–36.
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Philip, J. Plane sections of simplices. Mathematical Programming 3, 312–325 (1972). https://doi.org/10.1007/BF01585004
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DOI: https://doi.org/10.1007/BF01585004