# Analogue of subcontractive property of Hilbert space projections for general Banach spaces

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 $C$.

• 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". – Kavi Rama Murthy Jun 25 '18 at 6:18
• Very helpful. Thank you! – user572048 Jul 1 '18 at 22:12