Góźdź, Stanisław A convex polygon as a discrete plane curve. (English) Zbl 0951.52013 Balkan J. Geom. Appl. 4, No. 2, 29-48 (1999). The perimeter \(2\pi r\) of a circle with radius \(r\) can be obtained as the limit of perimeters of well-shaped regular polygons circumscribed on the circle. This idea and the counterpart of the Barbier theorem suggest that the perimeter \(\pi d\) for an oval with constant width \(d\) can be obtained in a similar way. To reach this aim the following theorems are proved: Every \(2n\)-polygon circumscribed on an oval with constant width \(\delta\) is a \(2n\)-polygon with constant diagonal equal to \(\delta/\cos\frac{\pi}{2n}.\) Every \(mn\)-polygon circumscribed on an oval with constant perimeter \(l\) of a circumscribed \(m\)-polygon is \(mn\)-polygon with constant perimeter \(l/\cos\frac{\pi}{mn}\) of a \(b\)-circumscribed \(m\)-polygon. For the proofs of these theorems, a discrete Fourier series of a periodic sequence is applied. Reviewer: Serguey M.Pokas (Odessa) MSC: 52C10 Erdős problems and related topics of discrete geometry 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) Keywords:convex polygons; \(b\)-circumscribed \(n\)-polygon; \(n\)-polygon with constant diagonal; \(2n\)-polygon circumscribed on an oval with constant width; \(m\)-polygon circumscribed on an \(mn\)-polygon; oval with constant perimeter of a circumscribe PDFBibTeX XMLCite \textit{S. Góźdź}, Balkan J. Geom. Appl. 4, No. 2, 29--48 (1999; Zbl 0951.52013) Full Text: EuDML EMIS