rwg>This exposes a bug in the terminology "unit ball", which really ought to mean "unit diameter ball". There is no local content maximum when the ball is inscribed in a unit cube.<rwg Is that right? Using diameter = 1 instead of radius = 1, I'm getting for A(d) / 2^(d-1) a maximum at d = 2.4765825823060852952076157688588232403016455151805+, and for V(d) / 2^d a maximum at d = 0.4765825823060852952076157688588232403016455151805+ . --Dan Yikes, I never considered 0<d<1, whereat the "sphere" exceeds the "bounding cube"! --rwg Question: Is that Amax-Vmax = 2 observation original? Did you use Area = d Volume/dr? --rwg DanA> Pretending that spheres, balls, and Euclidean spaces can have real dimensions: * let d_Amax := the real dimension d where the formula A(d) = 2 pi^(d/2) / Gamma(d/2) for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum. -and- * let d_Vmax := the real dimension d where the formula V(d) = pi^(d/2) / Gamma(d/2 + 1) for the d-dimensional content of the unit d-ball in R^d takes its maximum Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.