A sharp dual $$L_{p}$$ L p John ellipsoid problem for $$p\le -n-1$$ p ≤ - n - 1

Abstract

When $$p>0$$ , the dual $$L_p$$ John ellipsoids provide a unified treatment for the Lowner ellipsoid and the Legendre ellipsoid associated with a convex body. When $$p<0$$ , very little relevance to known ellipsoids has been found for the dual $$L_p$$ John ellipsoid so far. In this paper we investigate a sharp dual $$L_p$$ John ellipsoid problem associated with origin-symmetric convex bodies when $$p\le -n-1$$ . The solution unifies the classic John ellipsoid and the classic Petty ellipsoid.