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Proof.
⎛
⎝
⎞
⎠
Γ
n
π
n/
2
2
+1
r
n
lim
n→∞
·
n
n
n
+
2
≈
√
2
πe
−
⎛
⎝
⎞
⎠
c
re
√
π
)
n
n
n
+
2
c
n
n
n
+
2
=0
(
1
1
=
√
2
π
lim
n→∞
=
√
2
π
lim
n→∞
Corollary 2.
The fraction of the volume which lies at values of the radius be-
tween r
−
and r,where
0
<<ris
1
n
r
V
fraction
(
n, r,
)=1
−
−
Proof.
⎛
⎝
⎞
⎠
(
r−
)
n
·π
n/
2
Γ
(
2
+1
)
r
n
1
n
V
(
n, r
−
)
r
1
−
=1
−
=1
−
−
V
(
n, r
)
π
n/
2
Γ
(
2
+1
)
·
Corollary (1) implies that the higher the dimension the smaller the volume of a
hypersphere for a fixed radii. This property is investigated in more detail, in the
following section.
Corollary (2) reveals that in high-dimensional spaces, points which are uniformly
randomly distributed inside the hypersphere, are predominately concentrated in
a thin shell close to the surface or, in other words, at very high dimensions the
entire volume of a hypersphere is concentrated immediately below the surface.
Example 1.
Given a hypersphere with radius
r
=1
,
=0
.
1and
n
=50and
k
points which are uniformly randomly distributed inside the hypersphere, ap-
proximately 1
−
1
1
50
0
.
1
−
≈
99
,
5% of the
k
points lie within the thin
-shell
close to the surface.
4.1 Volume Extrema
By keeping the radius fixed and differentiating the volume
V
(
n, r
) with respect
to
n
, one obtains the dimension
4
where the volume is maximal :
r
n
r
n
π
n/
2
Ψ
2
+1
π
n/
2
Γ
2
+1
r
n
ln (
r
)
π
n/
2
Γ
2
+1
+
r
n
π
n/
2
ln (
π
)
∂
∂n
·
=
2
Γ
2
+1
−
2
Γ
2
+1
(3)
4
The dimension is obviously a nonnegative integer, however we consider term (3)
analytically as a real-valued function.
