Definition 3.113 . [02P9]

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Definition 3.113.

Let $Q_{i}$ , $i=0,\dots,n$ , be a family of compact convex subset of $M_{\mathbb{R}}$ and $g_{i}\colon Q_{i}\to\mathbb{R}$ a concave function on $Q_{i}$ . The mixed integral of $g_{0},\dots,g_{n}$ is defined as

\operatorname{MI}_{M}(g_{0},\dots,g_{n})=\sum_{j=0}^{n}(-1)^{n-j}\sum_{0\leq i_{0}<\cdots<i_{j}\leq n}\int_{Q_{i_{0}}+\cdots+Q_{i_{j}}}g_{i_{0}}\boxplus\cdots\boxplus g_{i_{j}}\,\text{\rm d}\operatorname{vol}_{M}.