On the Santaló-Yañez conjecture (Q1312281)

Let \(\{C(t)\}\) be a family of bounded closed convex sets in the hyperbolic plane \(H\), depending on the parameter \(t \geq 0\), such that \(C(t_ 1) \subset C(t_ 2)\) for \(t_ 1<t_ 2\) for \(t_ 1<t_ 2\), and for any point \(P \in H\), there is a \(t_ p\) such that \(P \in C(t)\) for each \(t \geq t_ p\). Let \(\sigma(C(t))\) and \(\ell (\partial C(t))\) denote the area of \(C(t)\) and the length of \(\partial C(t)\) respectively, and \(K<0\) be the curvature of \(H\). Then the Santaló-Yañez conjecture is that \[ \lim_{t \to \infty} {\ell (\partial C(t)) \over \sigma(C(t))}=(-K)^{1 \over 2}. \] The conjecture is not true for arbitrary convex sets. The authors present sufficient conditions for the conjecture to be true as well as some counterexamples.