2025/02/06 by Károly J. Böröczky, Boroczky, Karoly J., Ansgar Freyer +3
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Computational Geometry and Mesh Generation #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2502.04427
The paper focuses on possible hyperbolic versions of the classical Pal 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>2. 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 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 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 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 in Hn 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.