Pál’s Isominwidth Problem in the Hyperbolic Space

The paper focuses on possible hyperbolic versions of the classical Pál isominwidth inequality in R2 from 1921, which states that for a fixed minimal width, the regular triangle has minimal area. We note that the isominwidth problem is still wide open in Rn for n ≥ 3. Recent work on the isominwidth problem on the sphere S2 shows that the solution is the regular spherical triangle when the width is at most π2 according to Bezdek and Blekherman, while Freyer and Sagmeister proved that the minimizer is the polar of a spherical Reuleaux triangle when the minimal width is greater than π2 . In this paper, the hyperbolic isominwidth problem is discussed with respect to the probably most natural notion of width due to Lassak in the hyperbolic space Hn where strips bounded by a supporting hyperplane and a corresponding hypersphere are considered. On the one hand, we show that the volume of a convex body of given minimal Lassak width w > 0 in Hn might be arbitrarily small; therefore, the isominwidth problem for convex bodies inHn does not make sense. On the other hand, in the two-dimensional case, we prove that among horocyclically convex bodies of given Lassak width in H2, the area is minimized by the regular horocyclic triangle. In addition, we also verify a stability version of the last result.