  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.
