On a result by Meshulam

TL;DR

This paper extends Meshulam's 1996 result on boundedness of projection sequences from affine subspaces in Euclidean space to convex polyhedral subsets and general Hilbert spaces, with illustrative examples.

Contribution

It generalizes Meshulam's boundedness theorem from affine subspaces in Euclidean space to convex polyhedral sets and arbitrary Hilbert spaces.

Findings

Projection sequences onto convex polyhedral sets are bounded in Hilbert spaces.

The results are sharp, as demonstrated by various examples.

Abstract

In 1996, Meshulam proved that every sequence generated by applying projections onto affine subspaces, drawn from a finite collection in Euclidean space, must be bounded. In this paper, we extend his result not only from affine subspaces to convex polyhedral subsets, but also from Euclidean to general Hilbert space. Various examples are provided to illustrate the sharpness of the results.