Surrounding the solution of a linear system of equations from all sides

Suppose A ? R n × encoding="application/x-tex">(x_k)_{k=1}^{\infty } such all elements have same distance to alttext="x"> encoding="application/x-tex">x i.e. alttext="double-vertical-bar minus double-vertical-bar double-vertical-bar"> fence="false" stretchy="false">?<!-- ? <mml:mo>?<!-- ? encoding="application/x-tex">\|x_k - x\| \|x_1 x\| If chosen at random, averages over converge mathvariant="double-struck">E symmetric="true">? m ?<!-- ? </mml:munderover> ?<!-- ? <mml:mo>+ F ?<!-- ? <mml:mo>. encoding="application/x-tex">\begin{equation*} {E} \left \| \frac {1}{m} \sum _{k=1}^{m}{ x_k} \right \leq {1 + \|A\|_F \|A^{-1}\|}{\sqrt {m}} \cdot \|x-x_1\|. \end{equation*} The bound does not depend on dimension matrix. This introduces a purely geometric way attacking problem: there fast ways estimating location center sphere from knowing many points sphere? Our convergence rate (coinciding with Random Kaczmarz method) comes simple averaging, one do better?