Retracting a ball onto a sphere in some Banach spaces (Q608392)

The optimal retraction constant \(k_0(X)\) of a Banach space \(X\) is defined as the infimum of all \(k > 0\) such that there exists a retraction \(R: B_X \rightarrow S_X\) with Lipschitz constant \(k\), where \(B_X\) is the closed unit ball and \(S_X\) is the unit sphere. For the spaces \(C[0,1]\), \(BC(\mathbb{R})\), \(c_0\) and \(c\), the author proves that \[ k_0(X) \leq 4 (2 + \sqrt{3}). \]