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The title does not quite capture the essence of the difficulty, please allow me to be more explicit here. I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...