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I have an ellipse $\mathcal{E} = \{x^TAx = 1\}$, and I have a connected subset of an ellipse $U\subset \mathcal{E}$

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For a given $\theta$ let $x_U^*(\theta) = \arg \sup\{\langle x,\theta\rangle, x\in U\} $ and $x_\mathcal{E}^*(\theta) = \arg \sup\{\langle x,\theta\rangle, x\in \mathcal{E}\}$. Now in general $x_U^*(\theta) \neq x_\mathcal{E}^*(\theta)$.

But if $\theta$ and $U$ is such that $x_U^*(\theta)$ is an interior point of $U$ then it seems that both points are the same. This can be seen easily in the case of a sphere, though I cannot prove it. Can anyone help?

EDIT it's like suprema of a linear function over a convex surface occurs at boundary but if it is an interior point then it must be a global optima

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  • $\begingroup$ Ellipse has unique point with given outer normal vector $\theta$, thus any local maximum of $\langle x,\theta\rangle$ is a global maximum aswell. $\endgroup$
    – Fedor Petrov
    Commented May 6, 2022 at 16:56

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