Math Motivation: consider LINEAR subspace $L$ in $R^n$ and given vector $E$ in $R^n$, then it is easy to find a closest vector $S \in L$ to $E$ - just ortogonal projection.
Question Are they some interesting examples/constructions of non-linear manifolds/subsets $L$ in $R^n$ such that solve similar question for it is also "easy" ?
Well, "easy" means - not just direct use of some minimization algorithm...
Telecom motivation: set $L$ is set of signals which we want to "transmit", the map $L \to R^n$ is "error correcting coding" (i.e. adding redundant information), after the "transmission" due to noise we get point $E$ which might be out of the original set $L$. The "decoding" is the search of point $S$ in $L$ which is most close to received with errors point $E$.
So in the language of telecom theory my question is: how to build code which is "easy" to "decode". (At the moment I forget about the other important requirment - that code should correct as many errors as better)
There is clearly huge literature in coding theory. But may be some fresh look "ab initio" would be helpful (at least to clarify my mind).