The circumference of a convex polygon

1961/06/01 by F. Spitzer, Harold Widom, H. Widom · 5 citations
Mathematics · Engineering · #Mathematics and Applications #graph theory and CDMA systems #Functional Equations Stability Results

paper · doi:10.1090/s0002-9939-1961-0130616-7

Abstract

A CONVEX POLYGON f. spitzer and h.widom1In this note we combine a convexity theorem due to Cauchy with a combinatorial identity discovered by M. Kac.Cauchy's theorem [l] concerns the length L of the circumference of a compact convex set A in the plane.Let D(d) denote the projection of A on a line with direction 6, 0⁰<x, or, if z = x-\-iy(1) D(6) = max (x cos 6 + y sin 6) -min (x cos 0 + y sin 0). Then(2) L = f D(6)dd.Jo M. Kac, in [2], considered a vector x=(xi, x2, • • • , xn) with real components.For each permutation (.di cr2 ■ ■ • &n) he defined the vectors X(<r) \Xo Xaz, , Xa"), and their partial sums Jo(o-) = 0, Jfc(<T) =.,*#!+*r, + ••" + *»*» * = 1, 2, ■ • • , «.

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