Reference request for elementary convex geometry property

Question

I need to use the following lemma for a proof. It is an elementary result, which I am sure is well known, just I am not familiar enough with the relative literature to find a direct reference. Is there a name for this result? Or a simple reference for it? I would avoid writing the explicit proof if I could simply give a reference for it...

Lemma. Let $\mathcal X$ be a non-empty set in $\mathbb R^n$ and $C$ denote its convex hull. Let $x$ be a point in the relative boundary of $C$. Then, any Borel probability measure $P$, whose support lies in $\mathcal X$ and whose mean is $x$, must have its support included in the relative boundary of $C$.

The simplest proof I could think of is the following. For sure it can be shortened, but I'd rather avoid having to write it explicitly...

Proof. Let $x$ be in the relative boundary of $C$. Then, there is a supporting hyperplane $\Pi$ of $C$ that contains $x$ and does not intersect the relative interior of $C$ (and so, $\Pi\cap \mathcal X$ is included in the relative boundary). Now, by definition of supporting hyperplane, there is a non-zero vector $u$, orthogonal to $\Pi$, such that the mapping $F_u:y\mapsto y\cdot u$ is minimised by $x$ on $C$. Moreover, any other such minimiser $x'\in C$ must lie on $\Pi$ (and so must be in the relative boundary of $C$). Now, let $P$ be any Borel probability measure supported in $\mathcal X$ with mean $\langle P, X\rangle=x$ (as a side note, such $P$ must exist by Carathéodory theorem on convex hulls). Then, $\langle P, F_u(X)\rangle = u\cdot \langle P, X\rangle = F_u(x)$. Since $F_u(x')\geq F_u(x)$ for all $x'\in\mathcal X$, it must be that the support of $P$ is included in the set $\{x'\in\mathcal X\,:\,F_u(x)=F_u(x')\}$. But we had said earlier that such set must lie on $\Pi$ as all its elements are minimisers of $F_u$ on $C$. So the support of $P$ is included in $\Pi\cap\mathcal X$, which is in the relative boundary of $C$.

Answer 1

Indeed, this can be proved more simply, and in greater generality -- assuming only that the support of $P$ is contained in $C$ (rather than in $\mathcal X$).

Indeed, without loss of generality the affine hull of $C$ is $\Bbb R^n$. Then there is a nonzero linear functional $f$ such that $f(y)\le f(x)$ for all $y\in C$. Then the support of the pushforward $f_\# P$ of $P$ under $f$ is contained in the interval $(-\infty,f(x)]$ whereas the mean of $f_\# P$ is $f(x)$. So, the support of $f_\# P$ is $\{f(x)\}$; that is, the support of $P$ is contained in $C_{x,f}:=C\cap f^{-1}(\{f(x)\})$, and the set $C_{x,f}$ is contained in the relative boundary of $C$. $\quad\Box$

Since this proof is so simple (and your lemma is, at least intuitively, quite obvious), perhaps it does not exist as a separate statement in the literature.