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Let $ X $ be a Banach space over the reals with norm $ \| \cdot \| $. Let $ C $ be a nonempty closed convex subset of $ X $. Suppose $ a \notin C $. Is it true that there exists $ b \in C $ such that $ \|b - c\| < \|a - c\| $ for all $ c \in C $? Is it at least true in $ \mathbb{R}^d $ for $ \ell_p $ norms?

This property is clearly true for Hilbert spaces, since one can take $ b $ to be the projection of $ a $ onto $ C $.

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  • $\begingroup$ Rudin's Real and Complex Analysis has an example where there is no such $b$. See exercise 4 in the chapter on "Examples of Banach space Techniques". $\endgroup$ – Kavi Rama Murthy Jun 25 '18 at 6:18
  • $\begingroup$ Very helpful. Thank you! $\endgroup$ – user572048 Jul 1 '18 at 22:12

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