Polyak's theorem on Hilbert spaces
Functional Analysis
2022-02-10 v2
Abstract
We extend to infinite dimensional Hilbert spaces a celebrated result, due to B. Polyak, about the convexity of the joint image of quadratic functions. We give sufficient conditions which assure that the joint image is also closed. However, we show that, in general, the closedness part of Polyak's theorem does not hold in the infinite dimensional setting, even for quadratic functions generated by compact operators. We give some applications to S-lemma type results.
Cite
@article{arxiv.2107.10905,
title = {Polyak's theorem on Hilbert spaces},
author = {Maximiliano Contino and Guillermina Fongi and Santiago Muro},
journal= {arXiv preprint arXiv:2107.10905},
year = {2022}
}