Isodiametric problems for polygons (Q2505227)

The paper concerns two problems: finding convex \(n\)-gons of unit diameter which have possibly large area (possibly large perimeter). Recall that if \(n\) is odd, then the regular \(n\)-gon has the largest area [\textit{K. Reinhardt}, Extremale Polygone gegebener Durchmessers, Jahresber. Deutsch. Math.-Verein. 31, 251--270 (1922)]. For \(n\) even, the answer is known only for \(n=4\) (which is easy), for \(n=6\) [\textit{R. L. Graham}, J. Comb. Theory, Ser. A 18, 165--170 (1975; Zbl 0299.52006)], and for \(n=8\) [\textit{C. Audet, P. Hansen, F. Messine} and \textit{J. Xiong}, J. Comb. Theory, Ser. A 98, No.~1, 46--59 (2002; Zbl 1022.90013)]. The author of the present paper constructs a convex \(n\)-gon of unit diameter, where \(n\) is even, and shows that the area of this \(n\)-gon cannot be improved for large \(n\) by more than by \(cn^ {-3}\), where \(c\) is a constant. Moreover, for \(n=2^ m\) he presents a construction of a convex \(n\)-gon of unit diameter whose perimeter cannot be improved for large \(n\) by more than by \(cn^ {-5}\), where \(c\) is a constant.