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A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

8 votes
Accepted

Is a polytope that has in-spheres for faces of all dimensions already regular?

This is true in all dimensions, and can be proved by induction (on $d$) applied to the following (slightly stronger) hypothesis: Theorem: If $P$ is a convex $d$-polytope with $k$-in-spheres for all $ …
Adam P. Goucher's user avatar
5 votes
Accepted

Edges of the contact polytope of the Leech lattice

Using the unimodular scaling of the Leech lattice, the length of each minimal vector is $\sqrt{4}$. Fixing a particular minimal vector $u$, the remaining minimal vectors $v$ are: 1 vector $v$ with $\ …
Adam P. Goucher's user avatar