Let $X$ be a locally convex topological vector space, and let $f: X \times X \to [0,\infty]$.
Say that $f$ has the projection property if the following holds:
For all compact, convex $C \subseteq X$ and all $x \in X$, there exists a unique $p_x \in C$ (the "projection" of $x$ onto $C$) such that $f(p_x, x) \leq f(y, x)$ for all $y \in C$, and, moreover, if $x \in C$, then $p_x = x$.
I am wondering:
Are there some well-known conditions on $f$ that imply it has the projection property (e.g. $f$ is (strictly) convex and/or continuous in one/both of its arguments)? Can you point me to some references where the projection property and/or similar properties are studied?
If $f$ is continuous in its first argument, for example, then it achieves its minimum on every compact $C$. And if $f$ is also strictly convex in its first argument, then there is exactly one minimizer. But I don't see that these assumptions guarantee that the minimizer $p_x$ is $x$ whenever $x \in C$.
If possible, I would prefer not to assume that $X$ is metrizable.
One example of such a projection is the orthogonal projection. Notice that such $f$ is a function of $\mathbf{y}$, and $\mathbf{x}$ is merely a parameter. When the underlying space $\mathbf{X}$ is Euclidean, the orthogonal projection onto a set is defined as $$P_C(\mathbf{x})=\text{argmin}_{\mathbf{y}\in C}\|\mathbf{y}-\mathbf{x}\|.$$ Norm is a strictly convex function, but we require the domain to be convex in order to say convexity exists. Therefore when $C$ is a convex set, this $\text{argmin}$ exists and it is unique.
For such a projection, additional properties exist. Consider $\mathbf{y}\mapsto f(\mathbf{x}, \mathbf{y})=\text{argmin}_{\mathbf{y}\in C}\|\mathbf{y}-\mathbf{x}\|$, then following three claims are equivalent: \begin{align}&1)\; \mathbf{u} = P_C(\mathbf{x}) \\ &2)\; \mathbf{x}-\mathbf{u}\in \partial f(\mathbf{u})\\ &3)\; \langle\mathbf{x}-\mathbf{u},\mathbf{y}-\mathbf{u}\rangle\leq f(\mathbf{y}) -f(\mathbf{u}) & \forall \mathbf{y}\in \mathbf{X} \end{align} Another projection property is that $\mathbf{u}=P_C(\mathbf{x})$ if and only if $\langle\mathbf{x}-\mathbf{u},\mathbf{y}-\mathbf{u}\rangle\leq 0$ for all $\mathbf{y}\in C$.
Lastly, we have nonexpansivity and firm nonexpansivity. For all $\mathbf{x}, \mathbf{y}\in \mathbf{X}$: \begin{align}&1)\; \|P_C(\mathbf{x})-P_C(\mathbf{y})\|\leq\|\mathbf{x}-\mathbf{y}\| \\ &2)\; \|P_C(\mathbf{x})-P_C(\mathbf{y})\|^2\leq \langle\mathbf{x}-\mathbf{y}, P_C(\mathbf{x})-P_C(\mathbf{y})\rangle \end{align}