Is a closed set with the "unique nearest point" property convex?

It's true in Euclidean space, but not in general unless $K$ is weakly closed. You can see a proof for Euclidean space here


It is true for "finite-dimensional" Euclidean spaces and other special cases (e.g. real Hilbert space with boundedly compact subset $K$). However, this does not hold in general. The following paper provides a construction of a non-convex set with the nearest point property:

Johnson, Gordon G. "A nonconvex set which has the unique nearest point property." Journal of approximation theory 51.4 (1987): 289-332.

Hope this helps!

Tags:

Geometry

Hilbert Spaces

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