On the norm of the metric projections (Q1284489)

Let \(X\) be a Banach space and \(M\) a subspace of \(X\). Define \[ P_M(x)= \{y\in M:\| x-y\|= d(x,M)\}. \] The set-valued mapping \(P_M:X\to 2^M\), thus defined is called the metric projection onto \(M\). The space \(M\) is called proximinal if \(P_M(x)\neq \emptyset\) for every \(x\in X\). The norm of \(P_M\) is defined by \[ \| P_M\|= \sup\{\| y\|: y\in P_Mx\text{ and }\| x\|\leq 1\} \] and the metric constant \(\pi(X)\) is defined by \[ \pi(X)= \sup\{\| P_M\|: M\text{ a proximinal subspace of }X\}. \] The constant \(\mu(X)\) is defined as \[ \mu(X)= \sup\{d(E,\ell^2_2): E\subset X\text{ and }\dim E=2\}, \] where \(d(E,\ell^2_2)\) is the Banach-Mazur distance (the Banach-Mazur distance \(d(X,Y)\) between two isomorphic Banach spaces \(X\) and \(Y\) is defined by \(\inf\| T\| \| T^{-1}\|\), the infimum being taken over all invertible operators from \(X\) onto \(Y\)). The main results proved in the paper are: (i) For every Banach space \(X\), \(\pi(X)\leq (\mu(X))^2\), (ii) \(\pi(L_p)\leq 2^{| 2/p-1|}\), (iii) For every Banach space \(X\), \(\pi(X)= \pi(X^*)\).