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

Abstract

Abstract. For 0 < p ≤ ∞ and a convex body K in R n , Lutwak, Yang and Zhang defined the concept of dual Lp-centroid body Γ −p K and LpJohn ellipsoid EpK. 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 ≤ p ≤ 2 and 0 < q ≤ ∞,For 2 ≤ p ≤ ∞ and 0 < q ≤ ∞,(ii) For any convex body K whose John point is at the origin, there exists a simplex T such that for 1 ≤ p ≤ ∞ and 0 < q ≤ ∞,p EqT and V (K) = V (T ).