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arXiv · 1309.4534

Sufficiency of simplex inequalities

Shuzo Izumi
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Abstract

Let z_0,...,z_n be the (n-1)-dimensional volumes of facets of an n-simplex. Then we have the simplex inequalities: z_p < z_0+...+\check{z}_p+...+z_n (0 =< p =< n), generalizations of triangle inequalities. Conversely, suppose that numbers z_0,...,z_n > 0 satisfy these inequalities. Does there exist an n-simplex the volumes of whose facets are them? Kakeya solved this problem affirmatively in the case n = 3 and conjectured that the assertion is affirmative also for all n >= 4. We prove that his conjecture is affirmative. To do this, we define three kinds of spaces of loops associated to n-simplices and study relations among them systematically. In particular, we show that the space of edge loops corresponds to the space of facet loops bijectively under a certain condition of positivity.

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Shuzo Izumi. 2013-09-18. Sufficiency of simplex inequalities. https://arxiv.org/abs/1309.4534

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