Proposition 21.5.45D8A7A7D734E

Proof. Smooth functions. Let $F$ be smooth with positive definite Hessian. For $x\neq y$,

so $\nabla F$ is injective, and it is a local diffeomorphism by the inverse function theorem. Since $\partial F(E)=\nabla F(E)$ by proposition 21.2.4(3), the change of variables formula gives the asserted identity. Now let $F$ be smooth and convex; its Hessian is positive semidefinite, because the second derivatives of $F$ along lines are nonnegative. Apply the identity just proved to $F+\varepsilon |x|^2/2$, where $|x|$ is the Euclidean norm in the lattice coordinates. As $\varepsilon \downarrow 0$, these functions converge to $F$ locally uniformly and their densities $n!\det (D^2F+\varepsilon I)$ converge to $n!\det (D^2F)$ locally uniformly; comparing the two vague limits, by theorem 21.5.2, proves the identity for $F$.

Functions of class $C^{1,1}_{\mathrm{loc}}$. We work in the coordinates of a basis of $N$. Fix $k$ and $l$, and let $g=\partial _ kF$, which is locally Lipschitz. Let $B=B(x_0,r)$ be a ball, and let $L_ B$ be a Lipschitz constant of $g$ on $\overline{B(x_0,2r)}$. For a nonnegative test function $\chi $ supported in $B$ and $0<|h|<r$, the substitution $x\mapsto x-he_ l$ in one of the two terms gives

and the right side is at most $L_ B\int \chi \, \mathrm dx$ in absolute value. As $h\to 0$, the difference quotients of $\chi $ converge uniformly to $\partial _ l\chi $, with supports in a fixed compact set, so the left side tends to $\int g\, \partial _ l\chi \, \mathrm dx=-\langle \partial _ lg,\chi \rangle $. Hence $|\langle \partial _ l\partial _ kF,\chi \rangle |\leq L_ B\int \chi \, \mathrm dx$, and corollary 2.8.2(2), applied to Lebesgue measure, gives a Borel function $H_{kl}$ that represents the distribution $\partial _ l\partial _ kF$ and satisfies $|H_{kl}|\leq L_ B$ almost everywhere on each such ball $B$. Covering $N_{\mathbb R}$ by countably many such balls and setting $H_{kl}=0$ on the null set of points of some ball of the cover at which the bound fails, we may assume that $H_{kl}$ is bounded on every compact set. This proves that $D^2F=(H_{kl})$ is a locally bounded Borel matrix-valued function. Since distributional derivatives commute, $H_{kl}=H_{lk}$ almost everywhere, by the uniqueness in corollary 2.8.2.

Next, $D^2F(x)$ is positive semidefinite for almost every $x$. Indeed, let $v\in \mathbb R^ n$ and let $\chi $ be a nonnegative test function. Taylor's formula shows that the second difference quotients $h^{-2}(\chi (x+hv)+\chi (x-hv)-2\chi (x))$ converge uniformly to $\sum _{k,l}v_ kv_ l\partial _ k\partial _ l\chi $ as $h\to 0$, so the substitutions $x\mapsto x\mp hv$ give

because $F$ is convex. Thus the function $\sum _{k,l}v_ kv_ lH_{kl}$ defines a positive distribution, so its mollifications are nonnegative by proposition 2.5.3(3); since they converge to it at its Lebesgue points (theorem 2.8.3), it is nonnegative almost everywhere. Applying this to a countable dense set of vectors $v$, and using the continuity of the quadratic form in $v$, we see that $D^2F(x)$ is positive semidefinite for almost every $x$.

We now obtain the formula by mollification. With the mollifiers $\rho _\varepsilon $ of lemma 21.4.3, in the same coordinates, the functions $F_\varepsilon =F*\rho _\varepsilon $ are smooth and convex, and they converge to $F$ locally uniformly, by that lemma. By proposition 2.5.3(1), their second derivatives are $\partial _ k\partial _ lF_\varepsilon =H_{kl}*\rho _\varepsilon $. Hence, for $\varepsilon \leq 1$, the Hessians $D^2F_\varepsilon $ are bounded on each compact set $K$ by the bound of $D^2F$ on the set of points at distance at most $1$ from $K$, and by theorem 2.8.3 they converge to $D^2F(x)$ at every point $x$ that is a Lebesgue point of all the entries $H_{kl}$, that is, almost everywhere. By the smooth case, $\operatorname {MA}_{\mathbb R}(F_\varepsilon )=n!\det (D^2F_\varepsilon )\, \mathrm dx$. Let $\varepsilon $ run through a sequence tending to zero. For $f\in C_ c(N_{\mathbb R})$, dominated convergence gives $n!\int f\det (D^2F_\varepsilon )\, \mathrm dx\to n!\int f\det (D^2F)\, \mathrm dx$, while $\operatorname {MA}_{\mathbb R}(F_\varepsilon )\to \operatorname {MA}_{\mathbb R}(F)$ vaguely by theorem 21.5.2. This proves the formula.

Mixed measures. For smooth convex $F_1,\ldots ,F_ n$ define

where $D$ is the mixed discriminant of section 5.3, applied to the real symmetric, hence Hermitian, Hessian matrices. This measure is positive, because the mixed discriminant of positive semidefinite matrices is nonnegative (lemma 5.3.3(3)), and it is symmetric and multilinear, with $D(A,\ldots ,A)=\det A$ (lemma 5.3.3(1)). Expanding $\operatorname {MA}_{\mathbb R}(\sum _{j\in J}F_ j,\ldots ,\sum _{j\in J}F_ j)$ by multilinearity and using symmetry, we obtain the polarization identity

where the empty sum is the zero function, whose measure is zero for $n>0$ (its only slope is $0$). Indeed, after the expansion, a term $\operatorname {MA}_{\mathbb R}(F_{\sigma (1)},\ldots ,F_{\sigma (n)})$ with $\sigma \colon \{ 1,\ldots ,n\} \to \{ 1,\ldots ,n\} $ appears with total coefficient $\sum _{J\supset \sigma (\{ 1,\ldots ,n\} )}(-1)^{n-|J|}$, which vanishes unless $\sigma $ is a bijection, and the $n!$ bijections all give $\operatorname {MA}_{\mathbb R}(F_1,\ldots ,F_ n)$ by symmetry. Any symmetric, multilinear assignment with the given diagonal values satisfies (21.5.5), by the same expansion; this proves uniqueness.

Now let $F_1,\ldots ,F_ n$ be arbitrary finite convex functions, and let $F_{j,\varepsilon }=F_ j*\rho _\varepsilon $ be their mollifications, which are smooth and convex by lemma 21.4.3. For every $J$, $\sum _{j\in J}F_{j,\varepsilon }\to \sum _{j\in J}F_ j$ locally uniformly as $\varepsilon \to 0$ (lemma 2.5.1(1)). Let $\varepsilon $ run through a sequence tending to zero. By (21.5.5) for the smooth functions $F_{j,\varepsilon }$ and by theorem 21.5.2, for every $f\in C_ c(N_{\mathbb R},\mathbb R)$ the integrals $\int f\, \mathrm d\operatorname {MA}_{\mathbb R}(F_{1,\varepsilon },\ldots ,F_{n,\varepsilon })$ converge to

The functional $\Lambda $ is linear, and it is positive as a limit of integrals against positive measures. By theorem 2.3.3, there is a unique Radon measure $\operatorname {MA}_{\mathbb R}(F_1,\ldots ,F_ n)$ with $\int f\, \mathrm d\operatorname {MA}_{\mathbb R}(F_1,\ldots ,F_ n)=\Lambda (f)$ for every $f$; for smooth $F_ j$ it agrees with the previous definition, by (21.5.5). It is symmetric, because $\Lambda $ is a symmetric function of $F_1,\ldots ,F_ n$. Additivity and positive homogeneity in each argument hold for the smooth mixed measures; since mollification is linear, they pass to the vague limit. For $F_1=\cdots =F_ n=F$, the smooth mixed measures are $n!\det (D^2F_\varepsilon )\, \mathrm dx=\operatorname {MA}_{\mathbb R}(F_\varepsilon )$, which converge vaguely to $\operatorname {MA}_{\mathbb R}(F)$; hence $\operatorname {MA}_{\mathbb R}(F,\ldots ,F)=\operatorname {MA}_{\mathbb R}(F)$. Finally, if $F_ j^{(k)}\to F_ j$ locally uniformly as $k\to \infty $ for each $j$, then $\sum _{j\in J}F_ j^{(k)}\to \sum _{j\in J}F_ j$ locally uniformly for every $J$, and the formula for $\Lambda $ together with theorem 21.5.2 shows that $\operatorname {MA}_{\mathbb R}(F_1^{(k)},\ldots ,F_ n^{(k)})\to \operatorname {MA}_{\mathbb R}(F_1,\ldots ,F_ n)$ vaguely. In dimension zero the mixed measure has no arguments, and it is the unit mass at the point. $\square$