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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 $\xi_x$ ?

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  • $\begingroup$ According to your definition, the set of supporting vectors in the direction of $x$ (for $x \ne 0$) is a hyperplane. If $n\ge 3$ and $e_1$ and $e_2$ are linearly independent, the hyperplanes corresponding to $e_1$ and $e_2$ will always intersect, i.e. it is possible to have $\xi_{e_1} = \xi_{e_2}$. $\endgroup$
    – Robert Israel
    Commented Jan 21, 2020 at 15:37
  • $\begingroup$ Thank you for your remark, I forgot to mention that I choose $\xi_x\in\partial\Omega$ (and $\langle x,\xi_x\rangle=h(x)$). $\endgroup$
    – Buro
    Commented Jan 21, 2020 at 16:59

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