Yes, it is already known.

The story perhaps starts with:

Kostant's convexity theorem. Let $G/K$ be a symmetric space of compact type, $\mathfrak g=\mathfrak k +\mathfrak p$ the decomposition into the eigenspaces of the involution, $\mathfrak a$ a maximal Abelian subspace of $\mathfrak p$ and $W$ the restricted Weyl group, namely, the normalizer of $\mathfrak a$ in $K$ modulo its centralizer. Then, for any $a\in\mathfrak a$, the orthogonal projection of the adjoint orbit $\mathrm{Ad}_K(a)$ to $\mathfrak a$ (with respect to an $\mathrm{Ad}_K$-invariant inner product) is the convex polytope whose vertices are exactly the $w(a)$ for $w\in W$.

In particular, this theorem includes the case of adjoint or coadjoint orbits of a compact connected semisimple Lie group $G$, since $G$ with a bi-invariant Riemannian metric is the symmetric space $G\times G/\Delta_G$, where $\Delta_G$ is the diagonal subgroup. Coadjoint orbits $M$ are symplectic manifolds with respect to the Kirillov-Kostant-Souriau form, and the orthogonal projection onto the Lie algebra of the maximal torus $T$ is exactly the moment map of the $T$-action on $M$. A symplectic version of Kostant's theorem is then:

Atiyah-Guillemin-Sternberg's convexity theorem. Let $M$ be a compact symplectic manifold with a Hamiltonian action of a torus $T$. Then the image of the moment map $\mu:M\to\mathfrak t^*$ is the convex hull of the image under $\mu$ of the fixed point set of $T$ on $M$.

Orbits of isotropy representations of a symmetric spaces are isoparametric submanifolds of Euclidean space, namely, the eigenvalues of the shape operator along a locally defined parallel normal field are constant, and the normal bundle is flat. This provides for another generalization of Kostant's theorem:

Terng's convexity theorem. Let $M$ be a compact full isoparametric submanifold of Euclidean space and fix $p\in M$. Then the orthogonal projection of $M$ (or any parallel manifold through, say, $q\in p+\nu_pM$) to the normal space $p+\nu_pM$ is the convex hull of orbit of $p$ (resp. $q$) under the Weyl group $W$ associated to $M$.

Yes, it is already known.

The story perhaps starts with:

Kostant's convexity theorem. Let $G/K$ be a symmetric space of compact type, $\mathfrak g=\mathfrak k +\mathfrak p$ the decomposition into the eigenspaces of the involution, $\mathfrak a$ a maximal Abelian subspace of $\mathfrak p$ and $W$ the restricted Weyl group, namely, the normalizer of $\mathfrak a$ in $K$ modulo its centralizer. Then, for any $a\in\mathfrak a$, the orthogonal projection of the adjoint orbit $\mathrm{Ad}_K(a)$ to $\mathfrak a$ (with respect to an $\mathrm{Ad}_K$-invariant inner product) is the convex polytope whose vertices are exactly the $w(a)$ for $w\in W$.

In particular, this theorem includes the case of adjoint or coadjoint orbits of a compact connected semisimple Lie group $G$, since $G$ with a bi-invariant Riemannian metric is the symmetric space $G\times G/\Delta_G$, where $\Delta_G$ is the diagonal subgroup. Coadjoint orbits $M$ are symplectic manifolds with respect to the Kirillov-Kostant-Souriau form, and the orthogonal projection onto the Lie algebra of the maximal torus $T$ is exactly the moment map of the $T$-action on $M$. A symplectic version of Kostant's theorem is then:

Atiyah-Guillemin-Sternberg's convexity theorem. Let $M$ be a compact symplectic manifold with a Hamiltonian action of a torus $T$. Then the image of the moment map $\mu:M\to\mathfrak t^*$ is the convex hull of the image under $\mu$ of the fixed point set of $T$ on $M$.

Orbits of isotropy representations of a symmetric spaces are isoparametric submanifolds of Euclidean space, namely, the eigenvalues of the shape operator along a locally defined parallel normal field are constant, and the normal bundle is flat. This provides for another generalization of Kostant's theorem:

Terng's convexity theorem. Let $M$ be a compact full isoparametric submanifold of Euclidean space and fix $p\in M$. Then the orthogonal projection of $M$ (or any parallel manifold through, say, $q\in p+\nu_pM$) to the normal space $p+\nu_pM$ is the convex hull of orbit of $p$ (resp. $q$) under the Weyl group $W$ associated to $M$.

Reference: Convexity theorem for isoparametric submanifolds. Ch. L. Terng. Inventiones mathematicae 85 (1986), 487-492. (article)