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Results tagged with convex-polytopes
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user 4558
Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming
9
votes
1
answer
259
views
Do there exist "expanding" $1$-skeletons of simple $4$-polytopes?
Let $\{ G_n \}_{n \ge 1}$ be a sequence of graphs such that the number of vertices of $G_n$ tends to $\infty$ as $n \to \infty$. We say that $\{ G_n \}_{n \ge 1}$ is an expander family if
$\lambda_2( …
- 7,857
12
votes
On the number of Archimedean solids
Following up on Joseph's comment: Branko Grünbaum and others have pointed out that besides the 13 or 14, there are also two infinite families of polyhedra meeting the definition of Archimedean, altho …
- 7,857
33
votes
Open problems in Euclidean geometry?
The Unit Distance Problem asks:
For a set of $n$ points in the plane,
what is the maximal number $g(n)$ of
unit distances realized among the
${n \choose 2}$ pairs?
A properly scaled squar …