On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape

TL;DR

This paper investigates the minimum lengths of curves passing through boundary points of convex shapes, establishing bounds related to the shape's perimeter and exploring constraints involving extreme points.

Contribution

It proves that four boundary points always allow a curve of at least half the perimeter length, and analyzes limitations with extreme points.

Findings

Existence of four boundary points with curve length ≥ 1/2 perimeter

The half-perimeter bound does not hold with only extreme points

The factor 1/2 cannot be achieved with any fixed number of extreme points

Abstract

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 $\frac12$ cannot be achieved with any fixed number of extreme points. We conclude the paper with few other inequalities related to the perimeter of a convex shape.