An inequality concerning the Chebyshev ball of a bounded subset in a Hilbert space (Q1101924)

Let \(A\) be a non-empty bounded subset of a Hilbert space, with Chebyshev centre \(c\) and Chebyshev radius \(r(A)\). The author has proved the following inequality: \[ r(A)^ 2+\| x-c\| \leq F(x)^ 2, \] for each \(x\) in the closed ball \(B(c,r(A))\), where \(F(x)\) is the farthest distance of \(x\) from \(A\). Some fine applications with examples are also given.