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The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities

The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the $L_p$-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize $L_p$ spaces. For appropriate convex functions $φ:[0,\infty)^m\to [0,\infty)$, a new way of combining arbitrary sets in $\R^n$ is introduced. This operation, called Orlicz addition and denoted by $+_φ$, has several desirable properties, but is not associative unless it reduces to $L_p$ addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the first time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called $M$-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the $L_p$-Brunn-Minkowski theory. Inequalities of the Brunn-Minkowski type are obtained, both for $M$-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the $L_p$-Brunn-Minkowski inequality. New Orlicz-Minkowski inequalities are obtained that generalize the $L_p$-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.