Asymptotic behavior of convex sets in the hyperbolic plane (Q1075622)

We solve in the negative the following conjecture, posed by \textit{L. A. Santaló} and \textit{I. Yañez} in 1972 [J. Appl. Probab. 9, 140-157 (1972; Zbl 0231.60010)]: Let C(t) be a family of bounded closed sets in the hyperbolic plane, depending on the parameter t (0\(\leq t)\) and such that \(C(t_ 1)\subset C(t_ 2)\) for \(t_ 1<t_ 2\). Assume that for any point P of the plane there is a value \(t_ p\) of t such that, for all \(t\geq t_ p\), we have \(P\in C(t)\). If F(t) and L(t) denote respectively the area and length of C(t), prove that \(\lim_{t\to \infty}L(t)/F(t)=(-K)^{1/2}\) where \(K<0\) is the curvature of the hyperbolic plane. In fact we construct for each \(\lambda \in [(-K)^{1/2},\infty]\) a sequence of polygonal convex sets C(t) expanding over the whole hyperbolic plane with \(\lim_{t\to \infty}L(t)/F(t)=\lambda\). We also prove that the above sequence can be modified to obtain counterexamples with smooth boundary.