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Given two (low-dimensional, probably 2D) simplicial complexes P and Q, is there an efficient algorithm for constructing P', the subset of P consisting of all points in P which are the closest point to some point q in Q?

For instance, if P and Q were non-degenerately intersecting line segments, P' would be their intersection; if they were non-intersecting, P' would be a point or a segment. If P was a line segment and Q a triangle, P' would be the projection of Q onto P. If P was a triangle and Q was a line intersecting P, P' would consist of several incident line segments, from the interior and/or exterior of the triangle.

Some picture examples: (THE ONE WITH THE POINT INTERSECTION IS INCORRECT)

Examples of what I mean, described

In general P' seems to consist of the projections of Q onto each face (of any dimensionality) of P, but that description includes a large number of faces which are dominated by higher-dimensional faces, and it's not clear to me how to deal with that efficiently.

Sneftel
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    For the intersecting line segments case: can't P' be a line segment sometimes? E.g. in your diagram, pick a point somewhere on the green line Q and grow a circle around it until that circle touches the red line P: this point must be in P'. This suggests that P' will be the projection of Q onto P, just as in the triangle case. It seems to me that P' will only be a single point (the intersection) if P and Q are at right angles. – j_random_hacker Oct 14 '13 at 16:29
  • You're right, that example is wrong. – Sneftel Oct 14 '13 at 17:02

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