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Suppose a convex compact subset $X\subset \mathbb R^n$ is the intersection of all closed halfspaces that contain it, then with the inner product and the standard norm $\|.\|_2$,

$$X=\bigcap_{f\in \mathbb R^n, \|f\|_2\leq 1} \{ x\in \mathbb R^n : \langle f, x\rangle\leq \sup_{x\in X} \langle f,x\rangle\}$$

Since $f$ is used as a normal vector and only direction matters, can I replace $\|f\|_2\leq 1$ above equivalently by

$$\|f\|_{\infty}\leq 1$$

where $\|f\|_{\infty}=\max_{1\leq i\leq n} |f_i|$?

Thanks.

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Yes. It is important to note that the set on the right-hand side (within the intersection) does not depend on the length of the vector $f$. And for every vector $f \in \mathbb{R}^n \setminus \{0\}$, you have $$ \Bigg\| \frac{f}{\|f\|_2} \Bigg\|_2 = 1$$ and $$ \Bigg\| \frac{f}{\|f\|_\infty} \Bigg\|_\infty = 1.$$

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