The metric geometry of subspaces and convex cones of the Banach space revisited
A. B. Németh
TL;DR
This work presents a unified geometric framework for the metric projection problem in uniformly convex and uniformly smooth Banach spaces, tying linearity of projections on subspaces to convexity of polars of convex cones and to the inner product structure of the space. The core method leverages mutually polar retractions, Moreau decomposition, and the normalized duality mapping $J$ to translate subspace projection properties into polar-cone geometry; it shows that several natural conditions (on projections, dual maps, and polars) are equivalent and collectively characterize inner product spaces when $\dim X \ge 3$. The main theorem enumerates six equivalent statements, including linearity of projections on subspaces and on intersections of hyperplanes, the bidimensional-subspace behavior of $J^*$, polar convexity, and inner product structure, while highlighting the necessity of dimension at least three. Overall, the paper deepens the connection between projection geometry and convex analysis in Banach spaces and provides a precise, equivalence-based criterion for when a uniformly convex and smooth Banach space behaves like a Hilbert space.
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.