In linear algebra, we learn that we can project a vector $x \in \mathbb{R}^n$ onto a linear subspace $A \subseteq \mathbb{R}^n$. I have hunch that this can be generalized considerably. In particular, I think we should be able to replace $x$ and $A$ with arbitrary non-empty convex subsets of $\mathbb{R}^n$, call them $X$ and $A$.
In particular, define:
The vector from $X$ to $A$ is the unique $v$ in the closure of the difference set $A-X$ of minimum length. Such a $v$ always exists from Theorem 1.15 here. The length of this vector probably equals $\mathrm{inf}_{a \in A,x \in X}d(a,x),$ I guess.
The "projection" of $X$ onto $A$ is $X+v$; for sufficiently nice $A$ and $X$, this can be thought of as the minimum translation of $X$ so that it meets $A$.
More generally, this kind of thing should work in any sufficiently nice inner product space.
Question. It would be nice to have standard notation the aforementioned vector $v,$ and for the "projection" of $X$ onto $A$ as defined above. I'd also like some better terminology than "projection" because we're not actually flattening or changing the shape of anything; rather, we're just choosing a vector and translating by that vector.
Is anything approximately standard available?
Remark. This notion of projection is different to Wikipedia's notion, which is about finding elements in the intersection of two convex sets.