The metric geometry of subspaces and convex cones of the Banach space revisited

TL;DR

This paper establishes the equivalence between the linearity of metric projections, convexity of polars of convex cones, and the space being an inner product space in uniformly convex and smooth Banach spaces.

Contribution

It proves the equivalence of key geometric properties and inner product space structure in a specific class of Banach spaces.

Findings

Linearity of metric projections characterizes inner product spaces.

Convexity of polars of convex cones is equivalent to inner product space structure.

The results unify geometric properties in uniformly convex and smooth Banach spaces.

Abstract

It is proved that the linearity of metric projections on subspaces and the convexity of the polars of the convex cones in the uniformly convex and uniformly smooth Banach space are equivalent, and both of them is equivalent with the fact that the space is an inner product space.