  We show that, for vector spaces in which distance measurement is performed
using a gauge, the existence of best coapproximations in $1$-codimensional
closed linear subspaces implies in dimensions $\geq 2$ that the gauge is a
norm, and in dimensions $\geq 3$ that the gauge is even a Hilbert space norm.
We also show that coproximinality of all closed subspaces of a fixed dimension
implies coproximinality of all subspaces of all lower finite dimensions.
