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Minimal area ellipses in the hyperbolic plane. (English) Zbl 1291.52015
The departing point of this paper is a well-known result in convex geometry stating that a full-dimensional compact subset $$F$$ of the Euclidean plane can be enclosed by a unique ellipse $$C$$ of minimal area. Although the proof of this theorem is not difficult in the Euclidean plane, trying to extend it to compact subsets of the elliptic or hyperbolic plane presents a sufficient challenge. In their earlier paper [Adv. Geom. 12, No. 4, 665–684 (2012; Zbl 1291.51021)], the authors examined the situation in the elliptic plane and showed (using some complicated estimate techniques) that uniqueness can be guaranteed only for “sufficiently small and round sets $$F$$.”
In the paper under review, the authors prove an analogous result for compact sets in the hyperbolic plane, producing appropriate uniqueness results for enclosing ellipses of minimal area provided some sufficient conditions are met. Namely, an enclosing ellipse of minimal area (not assumed a priori unique) of a full-dimensional compact set $$F$$ in the hyperbolic plane is unique if there exist positive numbers $$\rho$$ and $$R$$ such that the semiaxes lengths of minimal ellipses are in the interval $$[\rho , \, R]$$ and the values $$v_1:= \coth^2R$$ and $$v_2:= \coth^2{\rho}$$ satisfy the inequality $$- 13v_1^2 + 5 v_1 v_2 - 3 v_1 + 7 v_2 + 4 \leq 0$$.
MSC:
 52A40 Inequalities and extremum problems involving convexity in convex geometry 52A55 Spherical and hyperbolic convexity 51M10 Hyperbolic and elliptic geometries (general) and generalizations 51M25 Length, area and volume in real or complex geometry 52A38 Length, area, volume and convex sets (aspects of convex geometry)
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