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...

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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)


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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).