zbMATH — the first resource for mathematics

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 \( - 13v_1^2 + 5 v_1 v_2 - 3 v_1 + 7 v_2 + 4 \leq 0\).
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)
Full Text: DOI arXiv Link
[1] Ball K.M.: Ellipsoids of maximal volume in convex bodies. Geom. Dedicata 41(2), 241–250 (1992) · Zbl 0747.52007
[2] Bastero J., Romance M.: John’s decomposition of the identity in the non-convex case. Positivity 6(1), 1–16 (2002) · Zbl 1018.52004
[3] Callahan J.J.: The Geometry of Spacetime. An Introduction to Special and General Relativity. Springer, New York (2000) · Zbl 0937.83001
[4] Davis C.: All convex invariant functions of Hermitian matrices. Arch. Math. 8(4), 276–278 (1957) · Zbl 0086.01702
[5] Gordon Y., Litvak A.E., Meyer M., Pajor A.: John’s decomposition of the identity in the general case and applications. J. Differ. Geom. 68(1), 99–119 (2004) · Zbl 1120.52004
[6] Gruber, P.M.: Results of Baire category type in convexity. In: Goodmann, J., Lutwak, E., Malkewitsch, E., Pollack, J. (eds.) Discrete Geometry and Convexity, pp. 163–169. New York Academy of Sciences (1985)
[7] Gruber, P.M.: Baire categories in convexity. In: Gruber, P.M., Wills, J. (eds.) Handbook of Convex Geometry, vol. B, pp. 1327–1346. Elsevier, Amsterdam (1993) · Zbl 0791.52002
[8] Gruber P.M.: Application of an idea of Voronoi to John type problems. Adv. Math. 218(2), 309–351 (2008) · Zbl 1144.52003
[9] Gruber, P.M.: John and Loewner ellipsoids. Discrete Comput. Geom. (2011). doi: 10.1007/s00454-011-9354-8 · Zbl 1241.52002
[10] Gruber P.M., Schuster F.E.: An arithmetic proof of John’s ellipsoid theorem. Arch. Math. 85, 82–88 (2005) · Zbl 1086.52002
[11] John, F.: Extremum problems with inequalities as subsidary conditions. In: Studies and Essays. Courant Anniversary Volume, pp. 187–204. Interscience, New York (1948)
[12] Lewis A.S.: Convex analysis on the Hermitian matrices. SIAM J. Optim. 6(1), 164–177 (1996) · Zbl 0849.15013
[13] Lutwak E., Yang D., Zhang G.: L p John ellipsoids. Proc. Lond. Math. Soc. 90, 497–520 (2005) · Zbl 1074.52005
[14] Özdemir M., Ergin A.A.: Rotations with unit timelike quaternions in Minkowski 3-space. J. Geom. Phys. 56(2), 322–336 (2006) · Zbl 1088.53010
[15] Reynold W.F.: Hyperbolic geometry on a hyperboloid. Am. Math. Monthly 100(5), 442–455 (1993) · Zbl 0789.51020
[16] Schröcker H.P.: Minimal enclosing hyperbolas of line sets. Beitr. Algebra Geom. 48(2), 367–381 (2007) · Zbl 1167.53003
[17] Schröcker H.P.: Uniqueness results for minimal enclosing ellipsoids. Comput. Aided Geom. Design 25(9), 756–762 (2008) · Zbl 1172.52300
[18] Weber M.J., Schröcker H.P.: Davis’ convexity theorem and extremal ellipsoids. Beitr. Algebra Geom. 51(1), 263–274 (2010) · Zbl 1202.52003
[19] Weber, M.J., Schröcker, H.P.: Minimal area conics in the elliptic plane. Adv. Geom. (2011, accepted). http://arxiv.org/abs/1008.4285 · Zbl 1291.51021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.