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Theorem 21.11.5859E2455286B4

Theorem 21.11.58.59E2455286B4 There is a symmetric map assigning to every $n$-tuple of finite convex functions $F_1,\ldots ,F_ n$ on $N_{\mathbb R}$ a positive Radon measure

\[ \operatorname{MA}_{\mathbb R}(F_1,\ldots ,F_ n) \]

on $N_{\mathbb R}$ with the following properties:

  1. CFA77F02E2A6

    For every $q\in \mathbb Z_{>0}$, finite convex functions $F_1,\ldots ,F_ q$ on $N_{\mathbb R}$, and $\lambda _1,\ldots ,\lambda _ q\geq 0$,

    \[ \operatorname{MA}_{\mathbb R}(\lambda _1F_1+\cdots +\lambda _ qF_ q) =\sum _{i_1,\ldots ,i_ n=1}^ q \lambda _{i_1}\cdots \lambda _{i_ n} \operatorname{MA}_{\mathbb R}(F_{i_1},\ldots ,F_{i_ n}). \]
  2. 72A14F74D8A4

    For every finite convex function $F$ on $N_{\mathbb R}$,

    \[ \operatorname{MA}_{\mathbb R}(F,\ldots ,F)=\operatorname{MA}_{\mathbb R}(F). \]
  3. 0430553123BF

    The map $\operatorname{MA}_{\mathbb R}$ is additive and positively homogeneous of degree $1$ in each entry. In other words, for finite convex functions $F,G,F_2,\ldots ,F_ n$ on $N_{\mathbb R}$ and $\lambda ,\mu \geq 0$,

    \[ \operatorname{MA}_{\mathbb R}(\lambda F+\mu G,F_2,\ldots ,F_ n) =\lambda \operatorname{MA}_{\mathbb R}(F,F_2,\ldots ,F_ n) +\mu \operatorname{MA}_{\mathbb R}(G,F_2,\ldots ,F_ n). \]

Proof. The construction of mixed Alexandrov measures gives a positive, symmetric, additive and positively homogeneous measure in the $n$ arguments, with diagonal equal to the Alexandrov measure defined here. Repeatedly expand the diagonal measure of $\sum_i\lambda_iF_i$ using this additivity and homogeneity. Each ordered choice $(i_1,\ldots,i_n)$ contributes $\lambda_{i_1}\cdots\lambda_{i_n}\operatorname{MA}_{\mathbb R}(F_{i_1},\ldots,F_{i_n})$, which is the asserted polynomial formula.

The earlier proof first verifies positivity for smooth convex functions using mixed determinants, and then uses convolution and local uniform continuity of the diagonal measure to pass to finite convex functions. In particular positivity is proved rather than inferred from a signed polarization formula. Uniqueness follows by recovering each coefficient from the diagonal polynomial, for instance by polarization. The normalization remains $n!$ on the diagonal.

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