Pál’s Isominwidth Problem in the Hyperbolic Space

Abstract The paper focuses on possible hyperbolic versions of the classical Pál isominwidth inequality in $${{\mathbb {R}}}^2$$ 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 $${{\mathbb {R}}}^n$$ R n for $$n\ge 3$$ n ≥ 3 . Recent work on the isominwidth problem on the sphere $$S^2$$ S 2 shows that the solution is the regular spherical triangle when the width is at most $$\frac{\pi }{2}$$ π 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 $$\frac{\pi }{2}$$ π 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 $${\mathbb {H}}^n$$ 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 $$w>0$$ w > 0 in $${\mathbb {H}}^n$$ H n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in $${\mathbb {H}}^n$$ 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 $${\mathbb {H}}^2$$ H 2 , the area is minimized by the regular horocyclic triangle. In addition, we also verify a stability version of the last result.