On evolutoids of planar convex curves
Let \(\gamma\) be a (regular) planar closed convex curve and consider a fixed angle \(\alpha\). The evolutoid \(\gamma_{\alpha}\) of \(\gamma\) is the envelope of lines making an angle \(\alpha\) with the tangent lines at \(\gamma\). Denote by \(A(\gamma)\) the area of the domain enclosed by \(\gamma\). The author proves the following: {\parindent=0.6 cm \begin{itemize}\item[(i)] if \(\gamma_{\alpha}\) is a convex curve, then \(A(\gamma_{\alpha}) \leq \cos^2(\alpha) A(\gamma)\), with equality if and only if \(\gamma\) is a Euclidean circle; \item[(ii)] for \(\alpha \in (0,{\pi \over 2})\), the curve \(\gamma\) is a circle if and only if \(\gamma_{\alpha}\) is a circle. \end{itemize}}
scientific article; zbMATH DE number 6351390