The Euclidean log convex density theorem, proved by Gregory Chambers in 2015, asserts that in Euclidean space with a log convex density spheres about the origin are isoperimetric. We provide a partial extension to hyperbolic space in which volume and perimeter densities are related but different.
Di Giosia, Leonardo; Habib, Jahangir; Kenigsberg, Lea; Pittman, Dylanger; and Zhu, Weitao
"The Log Convex Density Conjecture in Hyperbolic Space,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 18:
1, Article 9.
Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/9