EXTREMUM PROPERTIES OF DUAL Lp-CENTROID BODY AND Lp-JOHN ELLIPSOID

Abstract

For <TEX>$0</TEX><TEX><</TEX><TEX>p{\leq}{\infty}$</TEX> and a convex body <TEX>$K$</TEX> in <TEX>$\mathbb{R}^n$</TEX>, Lutwak, Yang and Zhang defined the concept of dual <TEX>$L_p$</TEX>-centroid body <TEX>${\Gamma}_{-p}K$</TEX> and <TEX>$L_p$</TEX>-John ellipsoid <TEX>$E_pK$</TEX>. In this paper, we prove the following two results: (i) For any origin-symmetric convex body <TEX>$K$</TEX>, there exist an ellipsoid <TEX>$E$</TEX> and a parallelotope <TEX>$P$</TEX> such that for <TEX>$1{\leq}p{\leq}2$</TEX> and <TEX>$0</TEX><TEX><</TEX><TEX>q{\leq}{\infty}$</TEX>, <TEX>$E_qE{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$</TEX> and <TEX>$V(E)=V(K)=V(P)$</TEX>; For <TEX>$2{\leq}p{\leq}{\infty}$</TEX> and <TEX>$0</TEX><TEX><</TEX><TEX>q{\leq}{\infty}$</TEX>, <TEX>$2^{-1}{\omega_n}^{\frac{1}{n}}E_qE{\subseteq}{\Gamma}_{-p}K{\subseteq}{2\omega_n}^{-\frac{1}{n}}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$</TEX> and <TEX>$V(E)=V(K)=V(P)$</TEX>. (ii) For any convex body <TEX>$K$</TEX> whose John point is at the origin, there exists a simplex <TEX>$T$</TEX> such that for <TEX>$1{\leq}p{\leq}{\infty}$</TEX> and <TEX>$0</TEX><TEX><</TEX><TEX>q{\leq}{\infty}$</TEX>, <TEX>${\alpha}n(nc_{n-2,p})^{-\frac{1}{p}}E_qT{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qT$</TEX> and <TEX>$V(K)=V(T)$</TEX>.