An interesting property of the evolute. (English) Zbl 1144.53007

Escudero, Carlos A.; Reventós, Agustí

Am. Math. Mon. 114, No. 7, 623-628 (2007).

This paper starts with an inequality on the boundary curve \(C\) of a convex, compact set \(K\) of area \(F\) in \(\mathbb R^2\) stating that the integral over the function \(1/\kappa\) along \(C\) exceeds \(2 F\) (\(\kappa\) being the curvature of \(C\)). The authors significantly improve this statement: The difference between the integral and the value \(2 F\) is recognized as the area \(2 F_e\), \(F_e\) being the area of the evolute to \(C\).

Reviewer: Johann Lang (Graz)

MSC:

53A04

Curves in Euclidean and related spaces

Keywords:

curvature; area; evolute