The equal tangents property

TL;DR

This paper proves that if all points outside a convex surface have equal tangent segment lengths, then the surface must be a sphere, highlighting a unique geometric property of spheres in three dimensions.

Contribution

It establishes a new characterization of spheres based on the equal tangent segments property for convex surfaces in space.

Findings

In three dimensions, the set of points with equal tangent segments forms a sphere if it contains the convex surface.

In the plane, the equal tangent segments property does not imply the shape is a circle.

The property distinguishes spheres from other convex shapes in Euclidean space.

Abstract

Let M be a smooth strictly convex closed surface in space and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface containing M or a plane then M is an Euclidean sphere. Moreover, we shall see that the situation in the Euclidean plane is very different.