Minimal area conics in the elliptic plane. (English) Zbl 1291.51021

Weber, Matthias J.; Schröcker, Hans-Peter

Adv. Geom. 12, No. 4, 665-684 (2012).

It has been long known that every compact subset \(F\) of full dimension in the Euclidean plane can be enclosed by a unique ellipse of minimal area [F. Behrend, Math. Ann. 115, 379–411 (1938; Zbl 0018.17502; JFM 64.0731.05)]. This uniqueness result is a relatively easy corollary of the convexity property of the function that measures the ellipse’s size. The extension of this result to convex sets in the elliptic or hyperbolic plane, however, is more difficult undertaking, which yields uniqueness result for superscribed conics only when certain (sufficient) conditions are postulated.
Given a bounded compact and full-dimensional subset \(F\) of the elliptic plane, the authors prove that among all conics with two given axes or with a given center that contain \(F\) there exists exactly one of minimal area. When the general case is considered, without restrictions on the enclosing conics in terms of the center or the axes, the authors give some non-obvious sufficient conditions under which the enclosing conic of minimal area of a compact subset \(F\) of the elliptic plane is unique. Analogous results are formulated for minimal enclosing conics of line sets.

Reviewer: Ivko Dimitric (Lemont Furnace)

MSC:

51M10	Hyperbolic and elliptic geometries (general) and generalizations
52A40	Inequalities and extremum problems involving convexity in convex geometry

Keywords:

minimal conic; sphero-conic; spherical ellipse; covering cone; enclosing conic; elliptic geometry

Citations:

Zbl 0018.17502; JFM 64.0731.05