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

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.