On the degree of regularity of a convex polygon

Abstract: We propose a canonical measure for the degree of regularity of a convex polygon.

In an article on this topic [1], a team of three researchers in Barcelona offers criteria that such a measure should meet, and then, like the man who hopped on his horse and galloped off in all directions, offers diverse measures suggested by diverse properties of regular polygons (the properties of being equilateral, equiangular, and of possessing radial symmetry). Moreover, they restrict consideration to polygons having no straight angles.

What we will do herein is obtain a measure of the degree of regularity of a convex polygon that simultaneously takes into account the three cited properties of regular polygons, and that does not need the restriction against straight angles. We do so by generalizing the notion of the apothem, heretofore applicable only to regular polygons, the generalization taking on computational form via averaging via the generalized geometric mean.

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On the degree of regularity of a convex polygon