Discrete sets with minimal moment of inertia

We analyze the moment of inertia , relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if . We introduce the notion of quasi-discs and show that roundest sets are strongly-convex quasi-discs in the discrete sense. We use weakly unimodal partitions and an inequality for the radius to make a table of roundest discrete sets up to size 40. Surprisingly, it turns out that the radius of the smallest disc containing a roundest discrete set S is not necessarily the radius of S as a quasi-disc.