Some New Positions of Maximal Volume of Convex Bodies

In this paper, we extend and generalize several previous works on maximal volume positions of convex bodies. First, we analyze the maximal positive-definite image of one convex body inside another, and the resulting decomposition of the identity. We discuss continuity and differentiability of the mapping associating a body with its positive John position. We then introduce the saddle-John position of one body inside another, proving that it shares some of the properties possessed by the position of maximal volume, and explain how this can be used to improve volume ratio estimates. We investigate several examples in detail and compare these positions. Finally, we discuss the maximal intersection position of one body with respect to another, and show the existence of a natural decomposition of identity associated to this position, extending previous work which treated the case when one of the bodies is the Euclidean ball.