On Dec 23, 2014, at 8:15 PM, Daniel Asimov <asimov@msri.org <mailto:asimov@msri.org>> wrote:
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.
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
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+.