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Sections of the regular simplex – Volume formulas and estimates
Author(s) -
Dirksen Hauke
Publication year - 2017
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600109
Subject(s) - hyperplane , mathematics , simplex , centroid , combinatorics , intersection (aeronautics) , subspace topology , volume (thermodynamics) , section (typography) , upper and lower bounds , geometry , mathematical analysis , physics , quantum mechanics , advertising , engineering , business , aerospace engineering
We state a general formula to compute the volume of the intersection of the regular n ‐simplex with some k ‐dimensional subspace. It is known that for central hyperplanes the one through the centroid containing n − 1 vertices gives the maximal volume. We show that, for fixed small distances of a hyperplane to the centroid, the hyperplane containing n − 1 vertices is still volume maximizing. The proof also yields a new and short argument for the result on central sections. With the same technique we give a partial result for the minimal central hyperplane section. Finally, we obtain a bound for k ‐dimensional sections.