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

33 votes
3 answers
2k views

Polar body of a convex body that avoids a lattice

Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of $K …
Terry Tao's user avatar
  • 114k
11 votes
0 answers
706 views

John-type theorems: trading structure for accuracy?

Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that $$ B \subset TB' \subset \tau B$$ for …
Terry Tao's user avatar
  • 114k