  If $K$ is a compact, connected, simply connected Lie group, its based loop
group $\Omega K$ is endowed with a Hamiltonian $S^1 \times T$ action, where $T$
is a maximal torus of $K$. Atiyah and Pressley examined the image of $\Omega K$
under the moment map $\mu$, while Jeffrey and Mare examined the corresponding
image of the real locus $\Omega K^\tau$ for a compatible anti-symplectic
involution $\tau$. Both papers generalize well known results in finite
dimensions, specifically the Atiyah-Guillemin-Sternberg theorem, and
Duistermaat's convexity theorem. In the spirit of Kirwan's convexity theorem,
this paper aims to further generalize the two aforementioned results by
demonstrating convexity of $\Omega K$ and its real locus $\Omega K^\tau$ in the
full non-abelian regime, resulting from the Hamiltonian $S^1\times K$ action.
In particular, this is done by appealing to the Bruhat decomposition of the
algebraic (affine) Grassmannian, and appealing to the "highest weight polytope"
results for Borel-invariant varieties of Guillemin and Sjamaar and Goldberg.
