Th Lp John ellipsoids for general measures

Abstract

This thesis aims to develop the Lp John ellipsoids related to general measures. Our Lp John ellipsoids contain many well-known ellipsoids constructed from given convex bodies as special cases, including but not limited to the classical John ellipsoid, the Lp John ellipsoid, the Lutwak-Yang-Zhang ellipsoid, the Petty ellipsoid, etc. Let μ be an α-homogeneous measure on Rn for α > 0. Our Lp John ellipsoids for the general measure μ for p > 0 are defined as the solutions to the following optimization problem: max V(E) subject to Vμ,p(K,E)≥μ(K), E∈ε₀n where E₀n denotes the set of all origin-symmetric ellipsoids, K is a compact convex set in Rn containing the origin in its interior, V is the volume function, and Vμ,p(K,E) = 1/αμ(K) [integral of] Sn⁻¹ hᴾE(v)dSμ,p(K,v), with hE the support function of E and dSμ,p(K,v) the Lp-surface μ-area measure of K. In this thesis, for p > 0, we establish the existence and uniqueness of the Lp John ellipsoid for μ. A characterization of the Lp John ellipsoid for μ is obtained. We also investigate the case for p = 0, which is related to the logarithmic function. Besides, the inclusion for the Lp John ellipsoid for μ is provided. The convex bodies with identical John and Lp John ellipsoids for the general measure μ are characterized. Finally, we provide a study for another arguably more general family of Lp John ellipsoids, defined in a way similar to the one in (1) but with Vμ,p(K,E) replaced by [integral of] Sn⁻¹ hᴾE(v)dv(v) and with μ(K) replaced by v(Sn⁻¹), respectively.