3.7. Monge-Ampère measures [02NK]

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3.7. Monge-Ampère measures

Let $f\colon C\to\mathbb{R}$ be a concave function of class ${\mathcal{C}}^{2}$ on an open convex set $C\subset\mathbb{R}^{n}$ . Its Hessian matrix

\operatorname{Hess}(f)(u):=\left(\frac{\partial^{2}f}{\partial u_{i}\partial u_{j}}(u)\right)_{1\leq i,j\leq n}

is a non-positive definite matrix which quantifies the curvature of $f$ at the point $u$ . The real Monge-Ampère operator is defined as $(-1)^{n}$ times the determinant of this matrix. This notion can be extended as a measure to the case of an arbitrary concave function. A good reference for Monge-Ampère measures is [RT77].

Let $\mu$ be a Haar measure of $M_{\mathbb{R}}$ . Assume that we choose linear coordinates $(x_{1},\dots,x_{n})$ of $M_{\mathbb{R}}$ such that $\mu$ is the measure associated to the differential form $\omega=\,\text{\rm d}x_{1}\land\dots\land\,\text{\rm d}x_{n}$ and the orientation of $M_{\mathbb{R}}$ defined by this system of coordinates. Let $(u_{1},\dots,u_{n})$ be the dual coordinates of $N_{\mathbb{R}}$ .

Definition 3.92.

Let $f$ be a concave function on $N_{\mathbb{R}}$ . The real Monge-Ampère measure of $f$ with respect to $\mu$ is defined, for a Borel subset $E$ of $N_{\mathbb{R}}$ , as

{\mathcal{M}}_{\mu}(f)(E)=\mu(\partial f(E)).

It is a measure with support contained in ${\operatorname{dom}}(\partial f)$ . The correspondence $f\mapsto{\mathcal{M}}_{\mu}(f)$ is called the Monge-Ampère operator.

When the measure $\mu$ is clear from the context, we will drop it from the notation. Moreover, since we are not going to consider complex Monge-Ampère measures, we will simply call ${\mathcal{M}}_{\mu}(f)$ the Monge-Ampère measure of $f$ .

The total mass of ${\mathcal{M}}_{\mu}(f)$ is equal to $\mu(\operatorname{stab}(f))$ . In particular, when $\operatorname{stab}(f)$ is bounded, ${\mathcal{M}}_{\mu}(f)$ is a finite measure.

Proposition 3.93.

The Monge-Ampère measure is a continuous map from the space of concave functions with the topology defined by uniform convergence on compact sets to the space of $\sigma$ -finite measures on $N_{\mathbb{R}}$ with the weak topology.

Proof.

This is proved in [RT77, §3]. ∎

The two basic examples of Monge-Ampère measures that we are interested in are the ones associated to smooth functions and the ones associated to piecewise linear functions.

Proposition 3.94.

Let $C$ be an open convex set in $N_{\mathbb{R}}$ and $f\in{\mathcal{C}}^{2}(C)$ a concave function. Then

{\mathcal{M}}_{\mu}(f)=(-1)^{n}\det(\operatorname{Hess}(f))\,\text{\rm d}u_{1}\land\dots\land\,\text{\rm d}u_{n},

where the Hessian matrix is calculated with respect to the coordinates $(u_{1},\dots,u_{n})$ .

Proof.

This is [RT77, Proposition 3.4] ∎

By contrast, the Monge-Ampère measure of a piecewise affine concave function, is a discrete measure supported on the vertices of a polyhedral complex.

Proposition 3.95.

Let $f$ be a piecewise affine concave function on $N_{\mathbb{R}}$ and $(\Pi(f),\Pi(f^{\vee}))$ the dual pair of polyhedral complexes associated to $f$ . Denote by $\Lambda\mapsto\Lambda^{\ast}$ the correspondence $\mathcal{L}f$ . Then

{\mathcal{M}}_{\mu}(f)=\sum_{v\in\Pi(f)^{0}}\mu(\partial f(v))\delta_{v}=\sum_{v\in\Pi(f)^{0}}\mu(v^{\ast})\delta_{v}=\sum_{\Lambda\in\Pi(f^{\vee})^{n}}\mu(\Lambda)\delta_{\Lambda^{\ast}},

where $\delta_{v}$ is the Dirac measure supported on $v$ .

Proof.

This follows easily from the definition of ${\mathcal{M}}(f)$ and the properties of the Legendre correspondence of piecewise affine functions. ∎

Example 3.96.

Let $\Delta\subset M_{\mathbb{R}}$ be a polytope and $\Psi_{\Delta}$ its support function. Then

{\mathcal{M}}_{\mu}(\Psi_{\Delta})=\mu(\Delta)\delta_{0}.

The following relation between Monge-Ampère measure and Legendre-Fenchel duality is one of the key ingredients in the computation of the height of a toric variety. We will consider the $(n-1)$ -differential form on $N_{\mathbb{R}}$

\lambda=\sum_{i=1}^{n}(-1)^{i-1}x_{i}\,\text{\rm d}x_{1}\land\dots\land\widehat{\,\text{\rm d}x_{i}}\land\dots\land\,\text{\rm d}x_{n}.

It satisfies $\,\text{\rm d}\lambda=n\omega$ .

Theorem 3.97.

Let $f\colon N_{\mathbb{R}}\to\mathbb{R}$ be a closed concave function, such that $D:=\operatorname{stab}(f)$ is a compact convex set with piecewise smooth boundary $\partial D$ . Then

(3.98)

-n!\int_{N_{\mathbb{R}}}f{\mathcal{M}}_{\mu}(f)=(n+1)!\int_{D}f^{\vee}\,\text{\rm d}\mu-n!\int_{\partial D}f^{\vee}\lambda.

Proof.

If the measure of $D$ is zero then both sides of equation (3.98) are zero. Therefore, the theorem is trivially true in this case. Thus, we may assume that $D$ has non-empty interior. Since $\operatorname{stab}(f)$ is compact, the right-hand side of (3.98) is continuous with respect to uniform convergence of functions, thanks to Proposition 3.18. Moreover, Proposition 3.93 and the fact that ${\mathcal{M}}_{\mu}(f)$ is finite imply that the left-hand side is also continuous with respect to uniform convergence. By the compacity of $D$ , we can find a sequence of strictly concave smooth functions $(f_{n})_{n\geq 1}$ that converges uniformly to $f$ . Hence, we may assume that $f$ is smooth and strictly concave. In this case, the Legendre transform $\nabla f\colon N_{\mathbb{R}}\to D^{\circ}$ is a diffeomeorphism.

By the definition of the Monge-Ampère measure,

(3.99)

-n!\int_{N_{\mathbb{R}}}f{\mathcal{M}}_{\mu}(f)=-n!\int_{D}f((\nabla f)^{-1}x)\,\text{\rm d}\mu(x),

which, in particular, shows that the integral on the left is convergent for smooth strictly concave functions with compact stability set. Therefore, it is convergent for any concave function within the hypothesis of the theorem.

By the properties of the Legendre transform,

(3.100)

-f((\nabla f)^{-1}(x))=f^{\vee}(x)-\langle(\nabla f)^{-1}(x),x\rangle.

Moreover,

	$\displaystyle\,\text{\rm d}(f^{\vee}\lambda)(x)$	$\displaystyle=\,\text{\rm d}f^{\vee}\land\lambda(x)+f^{\vee}\,\text{\rm d}\lambda(x)$
		$\displaystyle=\langle\nabla f^{\vee}(x),x\rangle\omega+nf^{\vee}\omega$
(3.101)			$\displaystyle=\langle(\nabla f)^{-1}(x),x\rangle\omega+nf^{\vee}\omega$

The result is obtained by combining equations (3.99), (3.100) and (3.101) with Stokes’ theorem. ∎

We now particularize Theorem 3.97 to the case when the Haar measure comes from a lattice and the convex set is a lattice polytope of maximal dimension.

Definition 3.102.

Let $L$ be a lattice and set $L_{\mathbb{R}}=L\otimes\mathbb{R}$ . We denote by $\operatorname{vol}_{L}$ the Haar measure on $L_{\mathbb{R}}$ normalized so that $L$ has covolume $1$ .

Let $N$ be a lattice of $N_{\mathbb{R}}$ and set $M=N^{\vee}$ for its dual lattice. For a concave function $f$ , we denote by $\mathcal{M}_{M}(f)$ the Monge-Ampère measure with respect to the normalized Haar measure $\operatorname{vol}_{M}$ .

Notation 3.103.

Let $\Lambda$ be a rational polyhedron in $M_{\mathbb{R}}$ and $\operatorname{aff}(\Lambda)$ its affine hull. We denote by $L_{\Lambda}$ the linear subspace of $M_{\mathbb{R}}$ associated to $\operatorname{aff}(\Lambda)$ and by $M(\Lambda)$ the induced lattice $M\cap L_{\Lambda}$ . By definition, $\operatorname{vol}_{M(\Lambda)}$ is a measure on $L_{\Lambda}$ , and we will denote also by $\operatorname{vol}_{M(\Lambda)}$ the measure induced on $\operatorname{aff}(\Lambda)$ . If $v\in N_{\mathbb{R}}$ is orthogonal to $L_{\Lambda}$ , we define $\langle v,\Lambda\rangle=\langle v,x\rangle$ for any $x\in\Lambda$ . Furthermore, when $\dim(\Lambda)=n$ and $F$ is a facet of $\Lambda$ , we will denote by $v_{F}\in N$ the vector of minimal length that is orthogonal to $L_{F}$ and satisfies $\langle v_{F},F\rangle\leq\langle v_{F},x\rangle$ for each $x\in\Lambda$ . In other words, $v_{F}$ is the minimal inner integral orthogonal vector of $F$ as a facet of $\Lambda$ .

Corollary 3.104.

Let $f$ be a concave function on $N_{\mathbb{R}}$ such that $\Delta=\operatorname{stab}(f)$ is a lattice polytope of dimension $n$ . Then

-n!\int_{N_{\mathbb{R}}}f{\mathcal{M}}_{M}(f)=(n+1)!\int_{\Delta}f^{\vee}\,\text{\rm d}\operatorname{vol}_{M}+\sum_{F}\langle v_{F},F\rangle n!\int_{F}f^{\vee}\,\text{\rm d}\operatorname{vol}_{M(F)},

where the sum is over the facets $F$ of $\Delta$ .

Proof.

We choose $(m_{1},\dots,m_{n})$ a basis of $M$ such that $(m_{2},\dots,m_{n})$ is a basis of $M(F)$ and $m_{1}$ points to the exterior direction. Expressing $\lambda$ in this basis we obtain

\lambda|_{F}=-\langle v_{F},F\rangle\,\text{\rm d}\operatorname{vol}_{M(F)}.

The result then follows from Theorem 3.97. ∎

In §6, we will see that we can express the height of a toric variety in terms of integrals of the form $\int_{\Delta}f^{\vee}\,\text{\rm d}\operatorname{vol}_{M}$ as in the above result. In some situations, it will be useful to translate those integrals to integrals on $N_{\mathbb{R}}$ .

Let $f\colon N_{\mathbb{R}}\to\mathbb{R}$ be a concave function and $g\colon\operatorname{stab}(f)\to\mathbb{R}$ an integrable function. We consider the signed measure on $N_{\mathbb{R}}$ defined, for a Borel subset $E$ of $N_{\mathbb{R}}$ , as

{\mathcal{M}}_{M,g}(f)(E)=\int_{\partial f(E)}g\,\text{\rm d}\operatorname{vol}_{M}.

Clearly, ${\mathcal{M}}_{M,g}(f)$ is uniformly continuous with respect to ${\mathcal{M}}_{M}(f)$ . By the Radon-Nicodym theorem, there is a ${\mathcal{M}}_{M}(f)$ -measurable function, that we denote $g\circ\partial f$ , such that

(3.105)

\int_{E}g\circ\partial f\,{\mathcal{M}}_{M}(f)=\int_{E}{\mathcal{M}}_{M,g}(f)=\int_{\partial f(E)}g\,\text{\rm d}\operatorname{vol}_{M}.

Example 3.106.

When the function $f$ is differentiable or piecewise affine, the measurable function $f^{\vee}\circ\partial f$ can be made explicit.

(1)

Let $f\in{\mathcal{C}}^{2}(N_{\mathbb{R}})$ . Proposition 3.94 and the change of variables formula imply $g\circ\partial f=g\circ\nabla f$ . For the particular case when $g=f^{\vee}$ , Theorem 3.52(4) implies, for $u\in N_{\mathbb{R}}$ ,

$f^{\vee}\circ\partial f(u)=\langle\nabla f(u),u\rangle-f(u).$

(2)

Let $f$ a piecewise affine concave function on $N_{\mathbb{R}}$ . By Proposition 3.95, ${\mathcal{M}}_{M}(f)$ is supported in the finite set $\Pi(f)^{0}$ and so is ${\mathcal{M}}_{M,g}(f)$ . For $v\in\Pi(f)^{0}$ write $v^{*}\in\Pi(f^{\vee})^{n}$ for the dual polyhedron. Then $g\circ\partial f(v)=\frac{1}{\operatorname{vol}_{M}(v^{*})}\int_{v^{*}}g\,\text{\rm d}\operatorname{vol}_{M}$ , which implies

f^{\vee}\circ\partial f(v)=\frac{1}{\operatorname{vol}_{M}(v^{*})}\int_{v^{*}}\langle x,v\rangle\,\text{\rm d}\operatorname{vol}_{M}-f(v).

The function $f^{\vee}\circ\partial f$ is defined as a ${\mathcal{M}}_{M}(f)$ -measurable function. Therefore, only its values at the points $v\in\Pi(f)^{0}$ are well defined. Nevertheless, we can extend the function $f^{\vee}\circ\partial f$ to the whole $N_{\mathbb{R}}$ by writing

f^{\vee}\circ\partial f(u)=\frac{1}{\operatorname{vol}_{\mu}(\partial f(u))}\int_{\partial f(u)}\langle x,u\rangle\,\text{\rm d}\mu-f(u)

for any Haar measure $\mu$ on the affine space determined by $\partial f(u)$ .

The Monge-Ampère operator is homogeneous of degree $n$ . It can be turned into a multi-linear operator which takes $n$ concave functions as arguments.

Definition 3.107.

Let $f_{1},\dots,f_{n}$ be concave functions on $N_{\mathbb{R}}$ . The mixed Monge-Ampère measure is defined by the formula

{\mathcal{M}}_{M}(f_{1},\dots,f_{n})=\frac{1}{n!}\sum_{j=1}^{n}(-1)^{n-j}\sum_{1\leq i_{1}<\cdots<i_{j}\leq n}{\mathcal{M}}_{M}(f_{i_{1}}+\dots+f_{i_{j}}).

It is a measure on $N_{\mathbb{R}}$ .

This operator was introduced by Passare and Rullgård [PR04]. It is multi-linear and symmetric in the variables $f_{i}$ .

Proposition 3.108.

The mixed Monge-Ampère measure is a continuous map from the space of $n$ -tuples of concave functions with the topology defined by uniform convergence on compact sets to the space of $\sigma$ -finite measures on $N_{\mathbb{R}}$ with the weak topology.

Proof.

The general mixed case reduces to the unmixed case $f_{1}=\dots=f_{n}$ , which is Proposition 3.93. ∎

Definition 3.109.

The mixed volume of a family of compact convex sets $Q_{1},\dots,Q_{n}$ of $M_{\mathbb{R}}$ is defined as

(3.110)

\operatorname{MV}_{M}(Q_{1},\dots,Q_{n})=\sum_{j=1}^{n}(-1)^{n-j}\sum_{1\leq i_{1}<\cdots<i_{j}\leq n}\operatorname{vol}_{M}(Q_{i_{1}}+\cdots+Q_{i_{j}})

Since $\operatorname{MV}_{M}(Q,\dots,Q)=n!\,\operatorname{vol}_{M}(Q)$ , the mixed volume is a generalization of the volume of a convex body. The mixed volume is symmetric and linear in each variable $Q_{i}$ with respect to the Minkowski sum, and monotone with respect to inclusion [Ewa96, Chapter IV].

The next result generalizes [PR04, Proposition 3] and shows that the mixed Monge-Ampère measure can be defined in terms of mixed volumes if the effective domains of the functions overlap sufficiently.

Proposition 3.111.

Let $f_{1},\dots,f_{n}$ be concave functions such that $\operatorname{ri}({\operatorname{dom}}(f_{1}))\cap\dots\cap\operatorname{ri}({\operatorname{dom}}(f_{n}))\neq\emptyset$ and $E\subset N_{\mathbb{R}}$ a Borel subset. Then

{\mathcal{M}}_{M}(f_{1},\dots,f_{n})(E)=\frac{1}{n!}\operatorname{MV}_{M}(\partial f_{1}(E),\dots,\partial f_{n}(E)).

If $f_{1},\dots,f_{k}$ are piecewise affine, this formula holds under the weaker hypothesis ${\operatorname{dom}}(f_{1})\cap\dots\cap{\operatorname{dom}}(f_{k})\cap\operatorname{ri}({\operatorname{dom}}(f_{k+1}))\cap\dots\cap\operatorname{ri}({\operatorname{dom}}(f_{n}))\neq\emptyset$ .

Proof.

This follows from Proposition 3.43 and the definition of the mixed Monge-Ampère measures and of mixed volumes. ∎

In particular, this gives the total mass of the mixed Monge-Ampère measure.

Corollary 3.112.

In the setting of Proposition 3.111, we have

{\mathcal{M}}_{M}(f_{1},\dots,f_{n})(N_{\mathbb{R}})=\frac{1}{n!}\operatorname{MV}_{M}(\operatorname{stab}(f_{1}),\dots,\operatorname{stab}(f_{n})).

Proof.

This follows readily from the above proposition and (3.22). ∎

Following [PS08a], we introduce an extension of the notion of integral of a concave function.

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}.

For a compact convex subset $Q\subset M_{\mathbb{R}}$ and a concave function $g$ on $Q$ , we have $\operatorname{MI}_{M}(g,\dots,g)=(n+1)!\int_{Q}g\,\text{\rm d}\operatorname{vol}_{M}$ . The mixed integral is symmetric and additive in each variable $g_{i}$ with respect to the sup-convolution. For a scalar $\lambda\in\mathbb{R}_{\geq 0}$ , we have $\operatorname{MI}_{M}(\lambda g_{0},\dots,\lambda g_{n})=\lambda\operatorname{MI}_{M}(g_{0},\dots,g_{n})$ . We refer to [PS08a, PS08b] for the proofs and more information about this notion.