(2, q)-sine Ellipsoids and Related Volume Inequalities

Abstract

Let K be a convex body that contains the origin in its interior and Q be a star body about the origin, respectively. In this paper, a new (2, q)-sine ellipsoid Λ_2,q(K, Q) is defined, and the Pythagorean theorems between Λ_2,q(K, Q) and the (2, q)-John ellipsoid E_2,q(K, Q) are obtained. Moreover, volume inequalities for Λ_2,q(K, Q) are established. At last, it is showed that the (2, q)-sine ellipsoid Λ_2,q(K, Q) is of valuation property.

Keywords

(2, q)-John ellipsoid / (2, q)-sine ellipsoid / Pythagorean theorem / volume inequality / valuation

References

[1] Alonso-Gutiérrez D, Bastero J. Approaching the Kannan–Lovász–Simonovits and Variance Conjectures, 2015, Cham: Springer.

[2] Bo YF, Xiong G. Extremal problems for L_p surface areas and John ellipsoids. J. Math. Anal. Appl., 2019, 479(1): 1226-1243.

[3] Borell C. The Brunn–Minkowski inequality in Gauss space. Invent. Math., 1975, 30(2): 207-216.

[4] Gardner RJ. The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. (N.S.), 2002, 39(3): 355-405.

[5] Gardner RJ. Geometric Tomography, 2006, Second, New York: Cambridge University Press.

[6] Giannopulos AA, Milman V. Euclidean structure in finite dimensional normed spaces. Handbook of the Geometry of Banach Spaces, Vol. 1, 2001, Amsterdam: Elsevier, 707-779.

[7] Grötschel M, Lovász L, Schrijver A. Geometric Algorithms and Combinatorial Optimization, 1988, Berlin: Springer-Verlag.

[8] Haberl C., Ludwig M., A characterization of L_p intersection bodies. Int. Math. Res. Not., 2006, 2006: Art. ID 10548, 29 pp.

[9] Hu JQ, Xiong G. The logarithmic John ellipsoid. Geom. Dedicata, 2018, 197: 33-48.

[10] Hu JQ, Xiong G, Zou D. On mixed L_p John ellipsoids. Adv. Geom., 2019, 19(3): 297-312.

[11] John F. Extremum problems with inequalities as subsidiary conditions. Studies and Essays Presented to R. Courant on his 60th Birthday, 1948, New York: Interscience Publishers, 187-204.

[12] Li AJ, Huang QZ, Xi DM. New sine ellipsoids and related volume inequalities. Adv. Math., 2019, 353: 281-311.

[13] Lu XB, Xiong G. The L_p John ellipsoids for negative indices. Israel J. Math., 2023, 255(1): 155-176.

[14] Lutwak E, Yang D, Zhang GY. A new ellipsoid associated with convex bodies. Duke Math. J., 2000, 104(3): 375-390.

[15] Lutwak E, Yang D, Zhang GY. L_p John ellipsoids. Proc. London Math. Soc. (3), 2005, 90(2): 497-520.

[16] Lutwak E, Yang D, Zhang GY. Lp dual curvature measures. Adv. Math., 2018, 329: 85-132.

[17] Ma TY, Wu DH. Extremum properties for dual L_p John ellipsoid. J. Interdiscip. Math., 2019, 22: 39-52.

[18] Ma TY, Wu DH, Feng YB. (p, q)-John ellipsoids. J. Geom. Anal., 2021, 31(10): 9597-9632.

[19] Osserman R. The isoperimetric inequality. Bull. Amer. Math. Soc., 1978, 84(6): 1182-1238.

[20] Pan SL, Tang XY, Wang XY. A strengthened form of Gage’s isoperimetric inequality. Chinese Ann. Math. Ser. A, 2008, 29(3): 301-306.

[21] Pan SL, Xu HP. Stability of a reverse isoperimetric inequality. J. Math. Anal. Appl., 2009, 350(1): 348-353.

[22] Pan SL, Zhang H. A reverse isoperimetric inequality for convex plane curves. Beiträge Algebra Geom., 2007, 48(1): 303-308.

[23] Petty CM. Centroid surfaces. Pacific J. Math., 1961, 11: 1535-1547.

[24] Schneider R. Convex Bodies: The Brunn–Minkowski Theory, 2014, Second, Cambridge: Cambridge University Press.

[25] Yang YL, Wu WP. The reverse isoperimetric inequality for convex plane curves through a length-preserving flow. Arch. Math. (Basel), 2021, 116(1): 107-113.

[26] Zou D, Xiong G. Orlicz–John ellipsoids. Adv. Math., 2014, 265: 132-168.

[27] Zou D, Xiong G. Orlicz–Legendre ellipsoids. J. Geom. Anal., 2016, 26(3): 2474-2502.

[28] Zou D, Xiong G. Convex bodies with identical John and LYZ ellipsoids. Int. Math. Res. Not. IMRN, 2018, 2018(2): 470-491.

[29] Zou D., Xiong G., New affine inequalities and projection mean ellipsoids. Calc. Var. Partial Differential Equations, 2019, 58(2): Paper No. 44, 18 pp.