Let $\Omega\subset\mathbb{R}^n$ a compact strictly convex set containing $0$ in its interior and let $k\leq n$.

Given a vector $x\neq 0$ in $\mathbb{R}^n$ a supporting vector $\xi_x$ in the direction of $x$ is a vector satifying $h(x)=\langle x,\xi_x\rangle$ where $h(x):=\sup\{\langle x,u\rangle|u\in\Omega\}$ is the supporting function of $\Omega$.

Given $k$ linearly independent directions $e_1,...,e_k$, on what condition on $\Omega$ can we show that the supporting vectors in the corresponding directions $\xi_{e_1},...,\xi_{e_k}$ are linearly independent?

More generally, what are the properties of the map associating to a unit vector $x$, the corresponding support direction $\xi_x$ ?