Coproximinality of linear subspaces in generalized Minkowski spaces

TL;DR

This paper investigates the conditions under which the existence of best coapproximations in generalized Minkowski spaces implies that the gauge is a norm or a Hilbert space norm, revealing deep geometric properties.

Contribution

It establishes that coproximinality in certain subspaces forces the gauge to be a norm or Hilbert space norm, extending understanding of geometric structure in these spaces.

Findings

Best coapproximations in 1-codimensional subspaces imply the gauge is a norm.

In dimensions ≥3, the gauge must be a Hilbert space norm.

Coproximinality of all subspaces of a fixed dimension implies coproximinality of all lower finite-dimensional subspaces.

Abstract

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.