Discrete sets with minimal moment of inertia
We analyze the moment of inertia I(S), 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 I(S) < I(T). 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. Crown Copyright (c) 2008 Published by Elsevier B.V. All rights reserved.
discrete sets moment of inertia polyominoes lattice paths