The equal tangents property

Let M be a C-smooth strictly convex closed surface in R 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. 1 In the Euclidean plane Let K be a strictly convex body in the plane. The following fact is well known: if the two tangent segments to K from every point x 6∈ K have equal lengths then K is an Euclidean disc (see, for instance, [4] ). This statement is easily proved by elementary geometry. The result was extended to the case of Minkowski planes by S. Wu [9], and Z. Lángi [3] also gave a characterization of the ellipsoid among centrally symmetric convex bodies in terms of tangent segments of equal Minkowski length. In the Euclidean plane, one may obtain the same conclusion with considerably weaker assumptions. Namely, one has the following characterization of a circle in terms of equal tangent segments.