A generalization of Löwner-John’s ellipsoid theorem

Abstract

We address the following generalization $\mathbf {P}$ of the Löwner-John ellipsoid problem. Given a (non necessarily convex) compact set $\mathbf {K}\subset \mathbb {R}^n$ and an even integer $d\in \mathbb {N}$, find an homogeneous polynomial $g$ of degree $d$ such that $\mathbf {K}\subset \mathbf {G}:=\{\mathbf {x}:g(\mathbf {x})\le 1\}$ and $\mathbf {G}$ has minimum volume among all such sets. We show that $\mathbf {P}$ is a convex optimization problem even if neither $\mathbf {K}$ nor $\mathbf {G}$ are convex! We next show that $\mathbf {P}$ has a unique optimal solution and a characterization with at most ${n+d-1\atopwithdelims ()d}$ contacts points in $\mathbf {K}\cap \mathbf {G}$ is also provided. This is the analogue for $d>2$ of Löwner-John’s theorem in the quadratic case $d=2$, but importantly, we neither require the set $\mathbf {K}$ nor the sublevel set $\mathbf {G}$ to be convex. More generally, there is also an homogeneous polynomial $g$ of even degree $d$ and a point $\mathbf {a}\in \mathbb {R}^n$ such that $\mathbf {K}\subset \mathbf {G}_\mathbf {a}:=\{\mathbf {x}:g(\mathbf {x}-\mathbf {a})\le 1\}$ and $\mathbf {G}_\mathbf {a}$ has minimum volume among all such sets (but uniqueness is not guaranteed). Finally, we also outline a numerical scheme to approximate as closely as desired the optimal value and an optimal solution. It consists of solving a hierarchy of convex optimization problems with strictly convex objective function and Linear Matrix Inequality constraints.

Appendix

1.1 First-order KKT-optimality conditions

Consider the finite dimensional optimization problem:

$$\begin{aligned} \inf \,\{\,f(\mathbf {x}):\,\mathbf {A}\mathbf {x}\,=\,\mathbf {b};\,\mathbf {x}\in \,C\,\}, \end{aligned}$$

for some real matrix $\mathbf {A}\in \mathbb {R}^{m\times n}$, vector $\mathbf {b}\in \mathbb {R}^m$, some closed convex cone $C\subset \mathbb {R}^n$ (with dual cone $C^*=\{\,\mathbf {y}:\mathbf {y}^T\mathbf {x}\ge 0,\,\forall \mathbf {x}\,\in C\,\}$) and some convex and differentiable function $f$ with domain $D$. Suppose that $C$ has a nonempty interior $\mathrm{int}(C)$ and Slater’s condition holds, that is, there exists $\mathbf {x}_0\in D\cap \mathrm{int}(C)$ such that $\mathbf {A}\mathbf {x}_0=\mathbf {b}$. The normal cone at a point $0\ne \mathbf {x}\in C$ is the set $N_C(\mathbf {x})=\{\mathbf {y}\in C^*:\,\langle \mathbf {y},\mathbf {x}\rangle =0\}$ (see e.g. [25, p. 189]).

Then by Theorem 5.3.3, p. 188 in [26], $\mathbf {x}^*\in C$ is an optimal solution if and only if there exists $(\lambda ,\mathbf {y})\in \mathbb {R}^m\times N_C(\mathbf {x}^*)$ such that:

$$\begin{aligned} \mathbf {A}\mathbf {x}^*=\mathbf {b};\quad \nabla f(\mathbf {x}^*)+\mathbf {A}^T\lambda \,=\,\mathbf {y}\end{aligned}$$

(7.1)

and $\langle \mathbf {x}^*,\mathbf {y}\rangle =0$ follows because $\mathbf {y}\in N_C(\mathbf {x}^*)$.

1.2 Measures with finite support

We restate the following important result stated in [29, Theorem 1] and [1, Theorem 2.1.1, p. 39].

Theorem 7.1

([1, 29]) Let $f_1,\ldots ,f_N$ be real-valued Borel measurable functions on a measurable space $\Omega$ and let $\mu$ be a probability measure on $\Omega$ such that each $f_i$ is integrable with respect to $\mu$. Then there exists a probability $\nu$ with finite support in $\Omega$ and such that:

$$\begin{aligned} \int _\Omega f_i(\mathbf {x})\,\mu (d\mathbf {x})\,=\,\int _\Omega f_i(\mathbf {x})\,\nu (d\mathbf {x}),\quad i=1\ldots ,N. \end{aligned}$$

One can even attain that the support of $\nu$ has at most $N+1$ points.

In fact if $\mathcal {M}(\Omega )_+$ denotes the space of probability measures on $\Omega$, then the moment space

$$\begin{aligned} Y_N:=\left\{ \mathbf {y}\,=\,\left( \int _\Omega f_k(\mathbf {x})d\mu (\mathbf {x})\right) ,\,k=1,\ldots ,N,\quad \text{ for } \text{ some } \mu \in \mathcal {M}(\Omega )_+\right\} \end{aligned}$$

is the convex hull of the set $f(\Omega )=\{(f_1(\mathbf {x}),\ldots ,f_N(\mathbf {x})):\,\mathbf {x}\in \Omega \}$ and each point $\mathbf {y}\in Y_N$ can be represented as the convex hull of at most $N+1$ point $f(\mathbf {x}_i)$, $i=1,\ldots ,N+1$. (See e.g. Sect. 3, p. 29 in Kemperman [28].)

In the proof of Theorem 3.2 one uses Theorem 7.1 with the $f_i$’s being all monomials $(\mathbf {x}^\alpha )$ of degree equal to $d$ (and so $N={n+d-1\atopwithdelims ()d}$). We could also use Tchakaloff’s Theorem [8] but then we would potentially need ${n+d\atopwithdelims ()d}$ points. An alternative would be to use Tchakaloff’s Theorem after “de-homogenizing” the measure $\mu$ so that $n$-dimensional moments of order $\vert \alpha \vert =d$ become $(n-1)$-dimensional moments of order $\vert \alpha \vert \le d$, and one retrieves the bound ${n-1+d\atopwithdelims ()d}$.

1.3 Proof of Theorem 3.2

Proof

• (a) As $\mathcal {P}$ is a minimization problem, its feasible set $\{\,g\in \mathbf {H}[\mathbf {x}]_{d}:1-g\in C_{d}(\mathbf {K})\,\}$ can be replaced by the smaller set

  $$\begin{aligned} F\,:=\,\left\{ g\in \mathbf {H}[\mathbf {x}]_{d}\,:\,\begin{array}{l}\displaystyle \int _{\mathbb {R}^n}\exp (-g(\mathbf {x}))\,d\mathbf {x}\,\le \,\int _{\mathbb {R}^n}\exp (-g_0(\mathbf {x}))\,d\mathbf {x}\\ 1-g\in \,C_{d}(\mathbf {K}) \end{array}\right\} , \end{aligned}$$

for some $g_0\in \mathbf {P}[\mathbf {x}]_{d}$. Notice that $F\subset \mathbf {P}[\mathbf {x}]_d$ and $F$ is a closed convex set since the convex function $g\mapsto \int _{\mathbb {R}^n}\exp (-g)d\mathbf {x}$ is continuous on the interior of its domain.

Next, let $\mathbf {z}=(z_\alpha ), \alpha \in \mathbb {N}^n_{d}$, be a (fixed) element of $\mathrm{int}(C_{d}(\mathbf {K})^*)$ (hence $z_0>0$). By Lemma 2.6 such an element exists and $\langle \mathbf {z},\mathbf {g}\rangle \ge 0$ (as $g\in \mathbf {P}[\mathbf {x}]_d$ is nonnegative). Next as $\langle \mathbf{z}, 1-\mathbf{g} \rangle \ge 0$ one has $z_0 \ge \langle \mathbf{z, g}\rangle$ and so by Corollary I.1.6 in Faraut and Korányi [15], g is bounded. Therefore the set $F$ is a compact convex set. Finally, since $g\mapsto \int _{\mathbb {R}^n}\exp (-g(\mathbf {x}))d\mathbf {x}$ is strictly convex, it is continuous on the interior of its domain and so it is continuous on $F$. Hence problem $\mathcal {P}$ has a unique optimal solution $g^*\in \mathbf {P}[\mathbf {x}]_d$.

• (b) We may and will consider any homogeneous polynomial $g$ as an element of $\mathbb {R}[\mathbf {x}]_{d}$ whose coefficient vector $\mathbf {g}=(g_\alpha )$ is such that $g^*_\alpha =0$ whenever $\vert \alpha \vert <d$. And so Problem $\mathcal {P}$ is equivalent to the problem

  $$\begin{aligned} \mathcal {P}':\quad \left\{ \begin{array}{ll}\rho =\displaystyle \inf _{g\in \mathbb {R}[\mathbf {x}]_{d}}&{}\displaystyle \int _{\mathbb {R}^n} \exp (-g(\mathbf {x}))\,d\mathbf {x}\\ \text{ s.t. }&{}g_\alpha =0,\quad \forall \,\alpha \in \mathbb {N}^n_{d};\,\vert \alpha \vert <d\\ &{}1-g\,\in \,C_{d}(\mathbf {K}),\end{array}\right. \end{aligned}$$

  (7.2)

  where we replaced $g\in \mathbf {P}[\mathbf {x}]_d$ with the equivalent constraints $g\in \mathbb {R}[\mathbf {x}]_{d}$ and $g_\alpha :=0$ for all $\alpha \in \mathbb {N}^n_{d}$ with $\vert \alpha \vert <d$. Next, doing the change of variable $h=1-g$, $\mathcal {P}$’ reads:

  $$\begin{aligned} \mathcal {P}':\quad \left\{ \begin{array}{ll}\rho =\displaystyle \inf _{h\in \mathbb {R}[\mathbf {x}]_{d}}&{}\displaystyle \int _{\mathbb {R}^n} \exp (h(\mathbf {x})-1)\,d\mathbf {x}\\ \text{ s.t. }&{}h_\alpha =0,\quad \forall \,\alpha \in \mathbb {N}^n_{d};\,0<\vert \alpha \vert <d\\ &{}h_0=1\\ &{}h\,\in \,C_{d}(\mathbf {K}),\end{array}\right. \end{aligned}$$

  (7.3)

As $\mathbf {K}$ is compact, there exists $\theta \in \mathbf {P}[\mathbf {x}]_{d}$ such that $1-\theta \in \mathrm{int}(C_{d}(\mathbf {K}))$, i.e., Slater’s condition holds for the convex optimization problem $\mathcal {P}'$. Indeed, choose $\mathbf {x}\mapsto \theta (\mathbf {x}):=M^{-1}\Vert \mathbf {x}\Vert ^{d}$ for $M>0$ sufficiently large so that $1-\theta >0$ on $\mathbf {K}$. Hence with $\Vert g\Vert _1$ denoting the $\ell _1$-norm of the coefficient vector of $g$ (in $\mathbb {R}[\mathbf {x}]_{d}$), there exists $\epsilon >0$ such that for every $h\in B(\theta ,\epsilon )(:=\{h\in \mathbb {R}[\mathbf {x}]_{d}:\Vert \theta -h\Vert _1<\epsilon \}$), the polynomial $1-h$ is (strictly) positive on $\mathbf {K}$.

Therefore, the unique optimal solution $(1-g^*)=:h^*\in \mathbb {R}[\mathbf {x}]_{d}$ of $\mathcal {P}$’ in (7.3) satisfies the Karush-Kuhn-Tucker (KKT) optimality conditions (7.1) which for problem (7.3) read:

$$\begin{aligned} \int _{\mathbb {R}^n}\mathbf {x}^\alpha \,\exp (h^*(\mathbf {x})-1)\,d\mathbf {x}&= y^*_\alpha , \quad \forall \vert \alpha \vert =d\end{aligned}$$

(7.4)

$$\begin{aligned} \int _{\mathbb {R}^n}\mathbf {x}^\alpha \,\exp (h^*(\mathbf {x})-1)\,d\mathbf {x}+ \gamma _\alpha&= y^*_\alpha ,\quad \forall \,\vert \alpha \vert <d\end{aligned}$$

(7.5)

$$\begin{aligned} \langle h^*,\mathbf {y}^*\rangle \,=\,0;\quad h^*_0\,=\,1;\,h^*_\alpha&= 0,\quad \forall \,0<\vert \alpha \vert <d \end{aligned}$$

(7.6)

for some $\mathbf {y}^*=(y^*_\alpha )$, $\alpha \in \mathbb {N}^n_{d}$, in the dual cone $C_{d}(\mathbf {K})^*\subset \mathbb {R}^{s(d)}$ of $C_{d}(\mathbf {K})$, and some vector $\gamma =(\gamma _\alpha )$, $0\le \vert \alpha \vert <d$. By Lemma 2.5,

$$\begin{aligned} C_{d}(\mathbf {K})^*\,=\,\left\{ \mathbf {y}\in \mathbb {R}^{s(d)}\,{:}\, \exists \mu \in \mathcal {M}(\mathbf {K})_+ \text{ s.t. } y_\alpha =\int _\mathbf {K}\mathbf {x}^\alpha \,d\mu ,\,\alpha \in \mathbb {N}^n_{d}\,\right\} , \end{aligned}$$

and so (3.2) is just (7.4) restated in terms of $\mu ^*$.

Next, the condition $\langle h^*,\mathbf {y}^*\rangle =0$ (or equivalently, $\langle 1-g^*,\mathbf {y}^*\rangle =0$), reads:

$$\begin{aligned} \int _{\mathbf {K}} (1-g^*)\,d\mu ^*\,=\,0, \end{aligned}$$

which combined with $1-g^*\in C_{d}(\mathbf {K})$ and $\mu ^*\in \mathcal {M}(\mathbf {K})_+$, implies that $\mu ^*$ is supported on $\mathbf {K}\cap \{\mathbf {x}:g^*(\mathbf {x})=1\}=\mathbf {K}\cap \mathbf {G}^*_1$.

Next, let $s:=\sum _{\vert \alpha \vert =d}g^*_\alpha y^*_\alpha \,(=y^*_0)$. From $\langle 1-g^*,\mu ^*\rangle =0$, the measure $s^{-1}\mu ^*=:\psi$ is a probability measure supported on $\mathbf {K}\cap \mathbf {G}^*_1$, and satisfies $\int \mathbf {x}^\alpha d\psi =s^{-1}y^*_\alpha$ for all $\vert \alpha \vert =d$ (and $\langle 1-g^*,\psi \rangle =0$).

Hence by Theorem 7.1 there exists an atomic probability measure $\nu ^*\in \mathcal {M}(\mathbf {K}\cap \mathbf {G}^*_1)_+$ such that

$$\begin{aligned} \int _{\mathbf {K}\cap \mathbf {G}^*_1}\mathbf {x}^\alpha d\nu ^*(\mathbf {x})\,=\,\int _{\mathbf {K}\cap \mathbf {G}^*_1}\mathbf {x}^\alpha d\psi (\mathbf {x})\,=\, s^{-1}\,y^*_\alpha ,\qquad \forall \,\vert \alpha \vert =d. \end{aligned}$$

In addition $\nu ^*$ may be chosen to be supported on at most $N={n+d-1\atopwithdelims ()d}$ points in $\mathbf {K}\cap \mathbf {G}^*_1$ and not $N+1$ points as predicted by Theorem 7.1. This is because one among the $N$ conditions

$$\begin{aligned} \int _{\mathbf {K}\cap \mathbf {G}_1^*}\mathbf {x}^\alpha \,d\nu ^*\,=\,s^{-1}\, y^*_\alpha ,\quad \vert \alpha \vert \,=\,d, \end{aligned}$$

is redundant as $\langle g^*,\mathbf {y}^*\rangle =y^*_0$ and $\nu ^*$ is supported on $\mathbf {K}\cap \mathbf {G}_1^*$. In other words, $\mathbf {y}^*$ is not in the interior of the moment space $Y_N$. Hence in (3.2) the measure $\mu ^*$ can be substituted with the atomic measure $s\,\nu ^*$ supported on at most $n+d-1\atopwithdelims ()d$ contact points in $\mathbf {K}\cap \mathbf {G}^*_1$.

To obtain $\mu ^*(\mathbf {K})=\frac{n}{d}\int _{\mathbb {R}^n}\exp (-g^*)$, multiply both sides of (7.4)-(7.5) by $h^*_\alpha$ for every $\alpha \ne 0$, sum up and use $\langle h^*,\mathbf {y}^*\rangle =0$ to obtain

$$\begin{aligned} -y^*_0=\sum _{\alpha \ne 0} h^*_\alpha \,y^*_\alpha&= \int _{\mathbb {R}^n}(h^*(\mathbf {x})-1)\exp (h^*(\mathbf {x})-1)\,d\mathbf {x}\\&= -\int _{\mathbb {R}^n}g^*(\mathbf {x})\exp (-g^*(\mathbf {x}))\,d\mathbf {x}\\&= -\frac{n}{d}\int \exp (-g^*(\mathbf {x}))\,d\mathbf {x}, \end{aligned}$$

where we have also used (2.5).

• (c) Let $\mu ^*:=\sum _{i=1}^s \lambda _i\delta _{\mathbf {x}_i}$ where $\delta _{\mathbf {x}_i}$ is the Dirac measure at the point $\mathbf {x}_i\in \mathbf {K}$, $i=1,\ldots ,s$. Next, let $y^*_\alpha :=\int \mathbf {x}^\alpha d\mu ^*$ for all $\alpha \in \mathbb {N}^n_{d}$, so that $\mathbf {y}^*\in C_d(\mathbf {K})^*$. In particular $\mathbf {y}^*$ and $g^*$ satisfy

  $$\begin{aligned} \langle 1-g^*,\mathbf {y}^*\rangle \,=\,\int _\mathbf {K}( 1-g^*)d\mu ^*\,=\,0, \end{aligned}$$

  because $g^*(\mathbf {x}_i)=1$ for all $i=1,\ldots ,s$. In other words, the pair $(g^*,\mathbf {y}^*)$ satisfies the KKT-optimality conditions associated with the convex problem $\mathcal {P}$. But since Slater’s condition holds for $\mathcal {P}$, those conditions are also sufficient for $g^*$ to be an optimal solution of $\mathcal {P}$, the desired result

$\square$

1.4 Proof of Theorem 4.1

Proof

First observe that (4.2) reads

$$\begin{aligned} \mathcal {P}:\quad \inf _{\mathbf {a}\in \mathbb {R}^n}\,\left\{ \inf _{g\in \mathbf {P}[\mathbf {x}]_d}\,\left\{ \mathrm{vol}(\mathbf {G}^\mathbf {a}_1)\,:\,1-g_\mathbf {a}\in C_d(\mathbf {K})\right\} \right\} , \end{aligned}$$

(7.7)

and notice that the constraint $1-g_\mathbf {a}\in C(\mathbf {K})$ is the same as $1-g\in C(\mathbf {K}-\mathbf {a})$. And so for every $\mathbf {a}\in \mathbb {R}^n$, the inner minimization problem

$$\begin{aligned} \inf _{g\in \mathbf {P}[\mathbf {x}]_d}\,\left\{ \mathrm{vol}(\mathbf {G}^\mathbf {a}_1)\,:\,1-g_\mathbf {a}\in C_d(\mathbf {K})\right\} \end{aligned}$$

of (7.7) reads

$$\begin{aligned} \rho _\mathbf {a}\,=\,\inf _{g\in \mathbf {P}[\mathbf {x}]_d}\,\left\{ \mathrm{vol}(\mathbf {G}_1)\,:\,1-g\in C_d(\mathbf {K}-\mathbf {a})\right\} . \end{aligned}$$

(7.8)

From Theorem 3.2 (with $\mathbf {K}-\mathbf {a}$ in lieu of $\mathbf {K}$), problem (7.8) has a unique minimizer $g^\mathbf {a}\in \mathbf {P}[\mathbf {x}]_d$ with value $\rho _\mathbf {a}=\int _{\mathbb {R}^n}\exp (-g^\mathbf {a})d\mathbf {x}=\int _{\mathbb {R}^n}\exp (-g^\mathbf {a}_\mathbf {a})d\mathbf {x}$.

Therefore, in a minimizing sequence $(\mathbf {a}_\ell ,g^{\mathbf {a}_\ell })\subset \mathbb {R}^n\times \mathbf {P}[\mathbf {x}]_d$, $\ell \in \mathbb {N}$, for problem $\mathcal {P}$ in (4.2) with

$$\begin{aligned} \rho =\lim _{\ell \rightarrow \infty }\,\int _{\mathbb {R}^n}\exp (-g^{\mathbf {a}_\ell })d\mathbf {x}, \end{aligned}$$

we may and will consider that for every $\ell$, the homogeneous polynomial $g^{\mathbf {a}_\ell }\in \mathbf {P}[\mathbf {x}]_d$) solves the inner minimization problem (7.8) with $\mathbf {a}_\ell$ fixed. For simplicity of notation rename $g^{\mathbf {a}_\ell }$ as $g^\ell$ and $g^{\mathbf {a}_\ell }_{\mathbf {a}_\ell }$ ($=g^{\mathbf {a}_\ell }(\mathbf {x}-\mathbf {a}_\ell )$) as $g^\ell _{\mathbf {a}_\ell }$.

As observed in the proof of Theorem 3.2, there is $\mathbf {z}\in \mathrm{int}(C_d(\mathbf {K})^*)$ such that $\langle 1-g^\ell _{\mathbf {a}_\ell },\mathbf {z}\rangle \ge 0$ and by Corollary I.1.6 in Faraut et Korányi [15], the set $\{h\in C_{d}(\mathbf {K}):\langle \mathbf {z},h\rangle \le z_0\}$ is compact.

Also, $\mathbf {a}_\ell$ can be chosen with $\Vert \mathbf {a}_\ell \Vert \le M$ for all $\ell$ (and some $M$), otherwise the constraint $1-g_{\mathbf {a}_\ell }\in C_d(\mathbf {K})$ would impose a much too large volume $\mathrm{vol}(\mathbf {G}^{\mathbf {a}_\ell }_1)$.

Therefore, there is a subsequence $(\ell _k)$, $k\in \mathbb {N}$, and a point $(\mathbf {a}^*,\theta ^*)\in \mathbb {R}^n\times C_d(\mathbf {K})$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\mathbf {a}_{\ell _k}\,=\,\mathbf {a}^*;\qquad \lim _{k\rightarrow \infty }(g^{\ell _k}_{\mathbf {a}_{\ell _k}})_\alpha \,=\,\theta ^*_\alpha ,\quad \forall \alpha \in \mathbb {N}^n_d. \end{aligned}$$

Recall the definition (4.1) of $g^\ell _{\mathbf {a}_\ell }(\mathbf {x})=g^\ell (\mathbf {x}-\mathbf {a}_\ell )$ for the homogeneous polynomial $g^\ell \in \mathbf {P}[\mathbf {x}]_d$ with coefficient vector $\mathbf {g}^\ell$, i.e.,

$$\begin{aligned} \left( g^\ell _{\mathbf {a}_{\ell }}\right) _\alpha \,=\,p_\alpha (\mathbf {a}_\ell ,\mathbf {g}^\ell ), \qquad \forall \alpha \in \mathbb {N}^n_d, \end{aligned}$$

for some polynomials $(p_\alpha )\subset \mathbb {R}[\mathbf {x},\mathbf {g}]$, $\alpha \in \mathbb {N}^n_d$. In particular, for every $\alpha \in \mathbb {N}^n_d$ with $\vert \alpha \vert =d$, $p_\alpha (\mathbf {a}_\ell ,\mathbf {g}^\ell )=(g^\ell )_\alpha$. And so for every $\alpha \in \mathbb {N}^n_d$ with $\vert \alpha \vert =d$,

$$\begin{aligned} \theta ^*_\alpha \,=\,\lim _{k\rightarrow \infty }\,=\,(g^{\ell _k})_\alpha . \end{aligned}$$

If we define the homogeneous polynomial $g^*$ of degree $d$ by $(g^*)_\alpha =\theta ^*_\alpha$ for every $\alpha \in \mathbb {N}^n_d$ with $\vert \alpha \vert =d$, then

$$\begin{aligned} \lim _{k\rightarrow \infty }\left( g^{\ell _k}_{\mathbf {a}_{\ell _k}}\right) _\alpha \,=\, \lim _{k\rightarrow \infty }p_\alpha \left( \mathbf {a}_{\ell _k},\mathbf {g}^{\ell _k}\right) , \,=\,p_\alpha (\mathbf {a}^*,\mathbf {g}^*),\quad \forall \alpha \in \mathbb {N}^n_d. \end{aligned}$$

This means that for every $\alpha \in \mathbb {N}^n_d$,

$$\begin{aligned} \theta ^*(\mathbf {x})\,=\,g^*(\mathbf {x}-\mathbf {a}^*),\quad \mathbf {x}\in \mathbb {R}^n. \end{aligned}$$

In addition, as $\mathbf {g}^{\ell _k}\rightarrow \mathbf {g}^*$ as $k\rightarrow \infty$, one has the pointwise convergence $g^{\ell _k}(\mathbf {x})\rightarrow g^*(\mathbf {x})$ for all $\mathbf {x}\in \mathbb {R}^n$. Therefore, by Fatou’s Lemma (see e.g. Ash [3]),

$$\begin{aligned} \rho \,=\,\lim _{k\rightarrow \infty }\,\int _{\mathbb {R}^n}\exp (-g^{\ell _k})\,d\mathbf {x}\,\ge \, \int _{\mathbb {R}^n}\liminf _{k\rightarrow \infty }\exp (-g^{\ell _k})\,d\mathbf {x}\,=\,\int _{\mathbb {R}^n}\exp (-g^*)\,d\mathbf {x}, \end{aligned}$$

which proves that $(\mathbf {a}^*,g^*)$ is an optimal solution of (4.2).

In addition $g^*\in \mathbf {P}[\mathbf {x}]_d$ is an optimal solution of the inner minimization problem in (7.8) with $\mathbf {a}:=\mathbf {a}^*$. Otherwise an optimal solution $h\in \mathbf {P}[\mathbf {x}]_d$ of (7.8) with $\mathbf {a}=\mathbf {a}^*$ would yield a solution $(\mathbf {a}^*,h)$ with associated cost $\int _{\mathbb {R}^n}\exp (-h)$ strictly smaller than $\rho$, a contradiction.

Hence by Theorem 3.2 (applied to problem (7.8)) there is a finite Borel measure $\mu ^*\in \mathcal {M}(\mathbf {K}-\mathbf {a}^*)_+$ such that

$$\begin{aligned} \int _{\mathbb {R}^n}\mathbf {x}^\alpha \exp (-g^*)d\mathbf {x}&= \int _{\mathbf {K}-\mathbf {a}^*} \mathbf {x}^\alpha \,d\mu ^*,\qquad \forall \vert \alpha \vert =d\\ \int _{\mathbf {K}-\mathbf {a}^*}(1-g^*)\,d\mu ^*&= 0;\quad \mu (\mathbf {K}-\mathbf {a}^*)=\frac{n}{d}\int _{\mathbb {R}^n}\exp (-g^*)\,d\mathbf {x}. \end{aligned}$$

And so $\mu ^*$ is supported on the set

$$\begin{aligned} V=\{\,\mathbf {x}\in \mathbf {K}-\mathbf {a}^*:\,g^*(\mathbf {x})=1\}\,=\,\{\,\mathbf {x}\in \mathbf {K}: \,g^*(\mathbf {x}-\mathbf {a}^*)=1\,\}\,=\,\mathbf {K}\cap \mathbf {G}^{\mathbf {a}^*}_1. \end{aligned}$$

Invoking again [1, Theorem 2.1.1, p. 39], there exists an atomic measure $\nu ^*\in \mathcal {M}(\mathbf {K}-\mathbf {a}^*)_+$ supported on at most ${n-1+d\atopwithdelims ()d}$ of $\mathbf {K}-\mathbf {a}^*$ with same moments of order $d$ as $\mu ^*$.   $\square$