Abstract
We prove that if a convex set in \({\mathbb {C}}^{n}\) contains two inscribed complex ellipsoids of maximal volume, then one is a translate of the other. On the other hand, the circumscribed complex ellipsoid of minimal volume is unique. As application we prove the complex analogue of Brunn’s classical characterization of ellipsoids.
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References
Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence (2002)
Bracho, Javier., Montejano, Luis.: On the complex Banach Conjecture. J. Convex Anal. To appear in Vol. 28 No.4 (2021)
Danzer, L., Laugwitz, D., Lenz, H.: About Löwner’s ellipsoid and its analogue among the ellipsoids inscribed in a convex body. Arch. Math. 8, 214–219 (1957). ((In German))
Gromov, M. L.: A geometrical conjecture of Banach, Izv. Akad. Nauk SSSR, Ser. Mat. Tom 31 (1967), No. 5:1055-1064
Heil, E., Martini, H.: Special convex bodies, in: Handbook of Convex Geometry, eds. P. M. Gruber and J. M. Wills, Vol. A, pp. 347-385, North-Holland, Amsterdam, 1993
Henk, Martin.: Löwner–John ellipsoids, Documenta Mathematica. Extra Volume ISMP: 95–106, 2012
Horn, Roger A., Johnson, Charles R.: Topics in Matrix Analysis. Cambridge University Press. (1991)
Martini, H., Montejano, L., Oliveros, D.: Bodies of Constant Width. An Introduction to Convex Geometry with Applications. Birkhäuser, Boston, Bassel, Stuttgart (2019)
Petty, C. M.,:Ellipsoids, in: Convexity and its Applications, Eds. P. M. Gruber and J. M. Wills, pp. 264–276, Birkhäuser, Basel, 1983
Zaguskin, V.L.: Circumscribed and inscribed ellipsoids of extremal volume. Usp. Mat. Nauk, 13(6 (84)): 89–93, 1958. (In Russian)
Funding
This work was supported by conacyt (Grant No. 166306), PAPIIT (Grant No. IN112614), PAPIIT (Grant No. IN109218).
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Arocha, J.L., Bracho, J. & Montejano, L. Extremal inscribed and circumscribed complex ellipsoids. Beitr Algebra Geom 63, 349–358 (2022). https://doi.org/10.1007/s13366-021-00608-w
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DOI: https://doi.org/10.1007/s13366-021-00608-w