PomeTheorem 4.5.6.B15EFD960FD5 Every nonempty closed convex subset $A$ of $\mathcal E^+$ has a unique nearest point $\mu _ A$ to $\mu \in \mathcal E^+$. It is characterized by $I(\mu -\mu _ A,\nu -\mu _ A)\le 0$ for all $\nu \in A$. For a closed convex cone with vertex zero this is equivalent to $I(\mu -\mu _ A,\nu )\le 0$ for all $\nu \in A$ and $I(\mu -\mu _ A,\mu _ A)=0$. For the cone of measures supported on compact $K$, the projection is exactly the balayage measure $\mu ^ K$: $G\mu ^ K=\widehat R_ K(G\mu )$. It is characterized by $G\mu ^ K\le G\mu $ everywhere and equality quasi-everywhere on $K$. The minimizer of $I(\nu )-2I(\mu ,\nu )$ on $A$ is this same projection. For nonpolar compact $K$, the unique probability of minimum energy is $\mu _ K/C(K)$ and its energy is $1/C(K)$.
Theorem 4.5.6B15EFD960FD5
Proof. The parallelogram identity bounds the distance between two almost minimizers by their errors above the infimum. Completeness of the positive cone and closedness of $A$ give the unique minimizing limit. Expanding the square along $\mu _ A+t(\nu -\mu _ A)$ and sending $t\downarrow 0$ gives the variational inequality; expanding the full square proves sufficiency. Testing zero and $2\mu _ A$ gives the cone characterization, with the converse immediate. Compact-support cones are closed because strong positive convergence implies vague convergence and preserves support.
The cone inequalities tested on restrictions of finite-energy measures on $K$ show $G\mu \le G\mu ^ K$ quasi-everywhere there; otherwise a positive-capacity compact subset of a level-set deficit supplies a finite-energy test with positive integral. Orthogonality then gives equality $\mu ^ K$-almost everywhere. Cartan domination gives $G\mu ^ K\le G\mu $ globally. Conversely these two pointwise conditions give all the cone inequalities, because energy measures charge no polar sets. The regularized reduction is a potential dominated by $G\mu $, has finite energy by potential order, and equals $G\mu $ quasi-everywhere on $K$; uniqueness identifies it with the projection. The affine functional differs from $\| \mu -\nu \| ^2$ by a constant. Finally $I(\mu _ K)=C(K)$ and $I(\mu _ K,\tau )=1$ for every finite-energy probability $\tau $ on $K$. Cauchy–Schwarz gives $I(\tau )\ge 1/C(K)$; equality and strict definiteness give uniquely $\tau =\mu _ K/C(K)$. Infinite-energy competitors cannot improve it. $\square$
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