Karoly Boroczky

Width and volume in the hyperbolic plane
(joint with Ansgar Freyer and Adam Sagmeister)

Inspired by the Kakeya problem, Pal showed in 1921 that in the Euclidean plane, the area of a convex domain with given minimal width is minimized by regular triangles. Recently, Pal's theorem has been generalized to the two-sphere, as well. The talk focuses on possible hyperbolic versions of Pal's theorem. In the hyperbolic world, even the notion of width is hard to clarify, and we use a recent natural notion of width due to Lassak. On the one hand, we show that the infimum of the area of convex domains of given Lassak width is zero. On the other hand, we prove that if horo-convex domains are considered (for any two points of a set X, the connecting two horocycle arcs are parts of X), then area is minimized by the "regular horo-triangle", that is bounded by three horocycle arcs.