Attainment and finite support for compact continuous mixtures

Define $\mathrm{\Phi }(y)=(y,a(y))\in {\mathbb{R}}^{d+1}$ and $G=\mathrm{\Phi }(K)$. The map $\mathrm{\Phi }$ is continuous. Each coordinate of $\mathrm{\Phi }$ is continuous on compact $K$, so [F4] bounds it; hence $G$ is bounded. To see $G$ is closed, take $(y,r)\mathrm{\notin }G$. If $y\mathrm{\notin }K$, closedness of $K$ gives a neighborhood of $y$ disjoint from $K$. If $y\in K$, then $r\mathrm{\ne }a(y)$, and continuity of $a$ gives neighborhoods of $y$ and $r$ whose product misses the graph of $a$. Thus the complement of $G$ is open. By [F3], $G$ is compact.

In a finite convex combination, discard zero-weight terms. If more than $d+2$ positive-weight points $g_i\in G\subset {\mathbb{R}}^{d+1}$ remain, the vectors $(g_i,1)\in {\mathbb{R}}^{d+2}$ are linearly dependent; hence there are coefficients $c_i$, not all zero, with ${\sum }_i{c}_i=0$ and ${\sum }_i{c}_i{g}_i=0$. Both signs occur. Replacing the positive weights ${\theta }_i$ by ${\theta }_i-t{c}_i$, where $t={\min }_{c_i>0}{\theta }_i\mathrm{/}{c}_i$, preserves their sum and combination, keeps them nonnegative, and makes at least one zero. Repeating gives at most $d+2$ points. Pad shorter combinations by zero weights and repeat any fixed point of nonempty $G$. The parameter set $G^{d+2}\times {\mathrm{\Delta }}_{d+2}$ is closed and bounded in a finite-dimensional Euclidean space, hence compact by [F3]. The map $\mathrm{\Theta }:G^{d+2}\times {\mathrm{\Delta }}_{d+2}\to {\mathbb{R}}^{d+1}$, $\mathrm{\Theta }((g_i),\theta )={\sum }_i{\theta }_i{g}_i$, is continuous and has image exactly $\mathrm{conv}G$: every open cover of the image pulls back to an open cover of the compact parameter set and therefore has a finite subcover. Thus $\mathrm{conv}G$ is compact and closed by [F3].

Fix $x\in \mathrm{conv}K$. Its fiber $F_x=\{r:(x,r)\in \mathrm{conv}G\}$ is nonempty, since each finite convex combination of generators gives such a point. It is closed and bounded because $\mathrm{conv}G$ is closed and bounded by [F3]; hence it is compact by [F3]. Therefore [F4] gives a minimum $r_x$. A convex-combination representation of $(x,r_x)$ by points $(y_i,a(y_i))$ defines a finite atomic Borel probability $P_x$ with barycenter $x$ and energy $r_x$.

Conversely, for any Borel probability $P$ on $K$, the vector-valued map $\mathrm{\Phi }$ is bounded and coordinatewise integrable by [F2]. Its integral is in $\mathrm{conv}G$: for each $ϵ>0$, cover compact $G$ by finitely many open Euclidean balls of radius $ϵ\mathrm{/}2$ and pull the cover back to $K$ by $\mathrm{\Phi }$. Disjointifying the finite preimages in order gives a finite Borel partition of $K$ whose pieces have image diameter at most $ϵ$. Choose a point in each nonempty piece and weight it by that piece's $P$-mass. The resulting finite convex combination differs coordinatewise from $\int \mathrm{\Phi }dP$ by at most $ϵ$; as $\mathrm{conv}G$ is closed by step 2.1, the integral belongs to it. If $\bar{y}(P)=x$, then $(x,\bar{a}(P))\in \mathrm{conv}G$, so $\bar{a}(P)\ge r_x$. Together with step 3.1 this proves that the all-measure infimum equals $r_x$ and is attained by $P_x$.

To prove lower semicontinuity, let $x_n\to x$ in $\mathrm{conv}K$ and put $L={\mathrm{lim\hspace{0.17em}inf}}_n{a}_{\mathrm{m}\mathrm{i}\mathrm{x}}(x_n)$. The values lie between ${\min }_{K}a$ and ${\max }_{K}a$, attained by [F4], so $L$ is finite. By the definition of liminf, choose a subsequence $n_j$ with $a_{\mathrm{m}\mathrm{i}\mathrm{x}}(x_{n_j})\to L$. Each pair $(x_{n_j},a_{\mathrm{m}\mathrm{i}\mathrm{x}}(x_{n_j}))$ lies in the closed set $\mathrm{conv}G$ by step 4.1, and the pairs converge to $(x,L)$; hence $(x,L)\in \mathrm{conv}G$. The minimum in the fiber at $x$ satisfies $a_{\mathrm{m}\mathrm{i}\mathrm{x}}(x)\le L={\mathrm{lim\hspace{0.17em}inf}}_n{a}_{\mathrm{m}\mathrm{i}\mathrm{x}}(x_n)$.

Combine minimizers $P_x$ and $P_z$ with weights $s$ and $1-s$, $0\le s\le 1$. The combined finite measure has barycenter $sx+(1-s)z$ and energy $s{a}_{\mathrm{m}\mathrm{i}\mathrm{x}}(x)+(1-s)a_{\mathrm{m}\mathrm{i}\mathrm{x}}(z)$. By step 4.1, the minimum at the combined barycenter is no greater than this value. Hence $a_{\mathrm{m}\mathrm{i}\mathrm{x}}$ is convex.

The support reduction in 2.1 first gives at most $d+2$ atoms. If an optimal representation has more than $d+1$ positive weights, the points $y_i\in {\mathbb{R}}^d$ have an affine dependence ${\sum }_i{c}_i{y}_i=0$, ${\sum }_i{c}_i=0$. For sufficiently small positive $t$, both weight lists ${\theta }_i\pm t{c}_i$ remain nonnegative and preserve the barycenter. If ${\sum }_i{c}_{i}a(y_i)\mathrm{\ne }0$, one of these lists strictly lowers energy, contradicting optimality; hence this sum is zero. Increase $t$ in one sign to the first value at which a positive coefficient becomes zero. The new weights preserve barycenter and energy and still attain the minimum; iteration leaves at most $d+1$ positive weights.

Finally, let $g$ be any convex function on $\mathrm{conv}K$ with $g(y)\le a(y)$ for $y\in K$. For the finite optimal measure from 4.1, finite Jensen (obtained by iterating the two-point convexity inequality) gives $g(x)\le {\sum }_i{\theta }_{i}g(y_i)\le {\sum }_i{\theta }_{i}a(y_i)=a_{\mathrm{m}\mathrm{i}\mathrm{x}}(x)$. Since $a_{\mathrm{m}\mathrm{i}\mathrm{x}}(y)\le a(y)$ for $y\in K$ by the point mass at $y$, it is itself such a convex minorant and dominates every other one.