Abstract The paper focuses on possible hyperbolic versions of the classical Pál isominwidth inequality in { {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 { {R}}ⁿ R n for n 3 n ≥ 3. Recent work on the isominwidth problem on the sphere S² S 2 shows that the solution is the regular spherical triangle when the width is at most 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 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 {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 {H}ⁿ H n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in {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 {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.
Böröczky et al. (Wed,) studied this question.