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We give a new characterisation of inner product spaces amongst normed vectorspaces in terms of the maximal cirumradius of spheres. It turns out that anormed vector space $(X,\norm{\cdot})$ with $\dim X\geq 2$ is an inner productspace if and only if all spheres are not degenerate, i.e. the maximalcircumradius of points on the sphere equals the radius of the sphere.

A characterisation of inner product spaces by the maximal circumradius of spheres