Given a convex body $K\in\mathbb{R}^n$, represented by a set of linear inequalities (intersection of halfspaces), I am interested in understanding how much of its volume can be close to its perimeter (under certain restrictions).

More formally, given a parameter $k$, and a partition of $\mathbb{R}^n$ into boxes of side $k$, I would like to know how large can the ratio $|P|/|I|$ be, where $P$ is the set of such boxes which intersect the perimeter, and $I$ is the set of boxes fully contained in $K$ (such a bound would be important for determining the accuracy of integration, for example).

I believe that as in 2 and 3 dimensions, the smallest ratio would be achieved by a ball (sphere), and that the worst ratio would be achieved for polytopes whose volume approaches 0 (by having a width smaller than k in one dimension, for example, which gives $|I|=0$).

However - are there some reasonable limits (for example, containing the 1/n unit sphere, or just having a volume > 0, and a poly(n) representation length of the linear inequalities) that can determine an upper bound on this ratio?

Many thanks, Guy

allof the boxes intersect the boundary, so your ratio is undefined. – Otis Chodosh Oct 28 '11 at 16:23