  Let $K$ be a compact Lie group and $V$ a finite-dimensional representation of
$K$. The orbitope of a vector $x\in V$ is the convex hull $\mathscr O_x$ of the
orbit $Kx$ in $V$. We show that if $V$ is polar then $\mathscr O_x$ is a
spectrahedron, and we produce an explicit linear matrix inequality
representation. We also consider the coorbitope $\mathscr O_x^o$, which is the
convex set polar to $\mathscr O_x$. We prove that $\mathscr O_x^o$ is the
convex hull of finitely many $K$-orbits, and we identify the cases in which
$\mathscr O_x^o$ is itself an orbitope. In these cases one has $\mathscr
O_x^o=c\cdot\mathscr O_x$ with $c>0$. Moreover we show that if $x$ has
"rational coefficients" then $\mathscr O_x^o$ is again a spectrahedron. This
provides many new families of doubly spectrahedral orbitopes. All polar
orbitopes that are derived from classical semisimple Lie can be described in
terms of conditions on singular values and Ky Fan matrix norms.
