Minimal area conics in the elliptic plane

Abstract

We prove uniqueness results for conies of minimal area that enclose a compact, fulldimensional subset of the elliptic plane. The minimal enclosing conic is unique if its center or axes are prescribed. Moreover, we provide sufficient conditions on the enclosed set that guarantee uniqueness without restrictions on the enclosing conies. Similar results are formulated for minimal enclosing conies of line sets as well.

Minimal area ellipses in the hyperbolic plane

“…The basic ideas and the initial calculations in our proof of Theorem 1 are more or less identical to the proof of [18,Theorem 8]. The minor differences pertain to occasional changes in sign and the use of the hyperbolic functions cosh, sinh, etc.…”

“…In [18] we used the spherical model of the elliptic plane for investigating uniqueness of minimal area conics. It is obtained from the geometry of the unit sphere S 2 of Euclidean three-space by identifying antipodal points.…”

“…Moreover, it follows from Lemma 3 and the strict version of Davis' convexity theorem [4,11] that area(C λ ) is a strictly convex function of λ. More detailed arguments can be found in [17,18]. The important fact to remember is that two enclosing conics C 0 and C 1 of the same area give rise to an in-between conic C λ of lesser area.…”

“…As usual, existence follows from compactness arguments. The basic ideas and initial steps in the proof of uniqueness are not different from the proof of Theorem 8 in [18]. We give an outline:…”

“…The general result (Theorem 1) involves rather cumbersome but straightforward calculations. The differences to the elliptic case are mainly in the final estimates for the coefficients of the polynomial P (t) in (18) and can be found in the appendix.…”