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This can be expanded in density as $h_{\\mathrm{KS}} \\propto n N [\\ln n a^d+ B + O(n a^d)+O(1/N)]$, with $a$ the diameter of the sphere or disk, $n$ the density, and $d$ the dimensionality of the system. We estimate the constant $B$ by solving a linear differential equation for the approximate distribution of eigenvalues of the inverse radius of curvature tensor. We compare the resulting values of $B$ both to previous estimates and to existing simulation results, finding very good agreement with the latter. 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