The paper focuses on possible hyperbolic versions of the classical Pál isominwidth inequality in \({{\mathbb {R}}}^2\) 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 \({{\mathbb {R}}}^n\) for \(n\ge 3\) . Recent work on the isominwidth problem on the sphere \(S^2\) shows that the solution is the regular spherical triangle when the width is at most \(\frac{\pi }{2}\) 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 \(\frac{\pi }{2}\) .
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 \({\mathbb {H}}^n\) 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 \({\mathbb {H}}^n\) might be arbitrarily small; therefore, the isominwidth problem for convex bodies in \({\mathbb {H}}^n\) 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 \({\mathbb {H}}^2\) , the area is minimized by the regular horocyclic triangle. In addition, we also verify a stability version of the last result.
