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The smallest dihedral angle of convex polyhedrons

Given a set of points $\{x_{k}\}_{k=0}^{m} \subset \mathbb{R}^n$, is it always possible to find a constant $c=c(m,n)>0$, depending only on the dimension $n$ and the number $m$, such that, after ...
18 votes
1 answer
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Sperner's Lemma implies Tucker's Lemma - simple combinatorial proof

Sperner’s Lemma is often called the "combinatorial analog" of Brouwer’s Fixed Point Theorem, and similarly Tucker’s Lemma is often called the combinatorial analog of Borsuk–Ulam’s Theorem. We can ...
2 votes
0 answers
82 views

Enveloping a Jordan curve with a trace of another one

This question is inspired by this one, or rather the way I understood it. Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\...
0 votes
1 answer
129 views

Fundamental Surfaces in 3-manifolds

Given a 3-manifold $M$ with a triangulation $T$, will every essential surface in $M$ be a fundamental one? If not, then what are the conditions on $T$ so that these essential surfaces become ...