Abstract
Let X be a real normed space with unit sphere S. It is proved that X is an inner product space if and only if there is a real number ρ, 0 < ρ < 1, and ρ ≠ ( 1 + cos ( 2 k π / n ) ) / 2 ( 2 k < n ; n = 3 , 4 , … ) , such that every chord of S that supports ρS touches ρS at its middle point.
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