On the lengths of curves passing through boundary points of a planar convex shape

article On the lengths of curves passing through boundary points of a planar convex shape published yes Arseniy Akopyan author 430D2C90-F248-11E8-B48F-1D18A9856A870000-0002-2548-617X Vladislav Vysotsky author HeEd department International IST Postdoc Fellowship Programme project We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor &#xbd; cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape. Mathematical Association of America2017 eng The American Mathematical Monthly 0002-9890 1605.07997 00041394730000210.4169/amer.math.monthly.124.7.588 1247588 - 596 Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” The American Mathematical Monthly, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96, doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a>. A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary points of a planar convex shape,” The American Mathematical Monthly, vol. 124, no. 7. Mathematical Association of America, pp. 588–596, 2017. Akopyan A, Vysotsky V. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 2017;124(7):588-596. doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a> A. Akopyan, V. Vysotsky, The American Mathematical Monthly 124 (2017) 588–596. Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596. Akopyan, A., & Vysotsky, V. (2017). On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. Mathematical Association of America. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a> Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” The American Mathematical Monthly. Mathematical Association of America, 2017. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>. 9092018-12-11T11:49:09Z2025-07-10T12:01:35Z