Bulletin of the
Korean Mathematical Society BKMS

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

Keywords: John ellipsoid, $L_{p}$-John ellipsoid, new ellipsoid, dual $L_{p}$-centroid body, simplex

MSC numbers: 52A40, 52A20