Title: On the Projection-Based Convexification of Some Spectral Sets

Abstract: Given a finite-dimensional real inner-product space E and a closed
convex cone K in R^n, we call $\lambda: E\to K$ a spectral map if  (E, R^n,
$\lambda$) forms a generalized Fan-Theobald-von Neumann (FTvN) system, and
call S a spectral set if $S := \lambda^{-1}(C)$ for some C in R^n. We
provide projection-based characterizations of clconv(S) (i.e., the closed
convex hull of S) under two settings, namely, when C has no invariance
property and when  C has certain invariance properties. In the former
setting, our approach is based on characterizing the bi-polar set of S,
which allows us to judiciously exploit the properties of $\lambda$ via
convex dualities. In the latter setting, our results complement the existing
characterization of clconv(S) in Jeong and Gowda (2023), and  unify and
extend the related results in Kim et al. (2022) established for certain
special cases of $\lambda$ and C.