On Minimal Moment of Inertia Polyominoes
We analyze the moment of inertia Open image in new window , 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 Open image in new window . We show that roundest sets of a given size are strongly convex in the discrete sense. Moreover, we introduce the notion of quasi-discs and show that roundest sets are quasi-discs. 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.
KeywordsDiscrete sets moment of inertia polyominoes lattice paths
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