The number of weakly compact convex subsets of the Hilbert space

TL;DR

This paper demonstrates that for any uncountable cardinal, there are exponentially many non-homeomorphic weakly compact convex subsets of a Hilbert space with the same weight and density, highlighting the vast diversity of such subsets.

Contribution

It establishes the existence of 2^k non-homeomorphic weakly compact convex subsets in Hilbert spaces for uncountable cardinals k, revealing a rich structural complexity.

Findings

Existence of 2^k non-homeomorphic weakly compact convex subsets for uncountable k

Diversity of convex subsets in Hilbert spaces at uncountable scales

Structural complexity of weakly compact convex sets in infinite-dimensional spaces

Abstract

We prove that for k an uncountable cardinal, there exist 2^k many non homeomorphic weakly compact convex subsets of weight k in the Hilbert space of density k.