TITLE: Sample integer points of cross-polytope uniformly
QUESTION [6 upvotes]: For $r,d\in\mathbb{N}$, let 
$$C_{r,d}=\{x\in\mathbb{Z}^d: \|x\|_1\le r\}\subset\mathbb{Z}^d$$ 
be the set of integer points of the $d$-dimensional cross-polytope with radius $r$. 
What is (currently) the fastest way to sample from $C_{r,d}$ uniformly? Is there a method which is polynomial in $d$ and $\log(r)$?

REPLY [2 votes]: Look at page two of Kannan and Vempala.