In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An -ball is a ball in an -dimensional Euclidean space. The volume of a -ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a -ball of radius is where is the volume of the unit -ball, the -ball of radius .
The real number can be expressed via a two-dimension recurrence relation. Closed-form expressions involve the gamma, factorial, or double factorial function. The volume can also be expressed in terms of , the area of the unit -sphere.
Formulas
The first volumes are as follows:
Closed form
The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:
where is Euler's gamma function. The gamma function is offset from but otherwise extends the factorial function to non-integer arguments. It satisfies if is a positive integer and if is a non-negative integer.
Two-dimension recurrence relation
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation:
This allows computation of in approximately steps.
Alternative forms
The volume can also be expressed in terms of an -ball using the one-dimension recurrence relation:
Inverting the above, the radius of an -ball of volume can be expressed recursively in terms of the radius of an - or -ball:
Using explicit formulas for particular values of the gamma function at the integers and half-integers gives formulas for the volume of a Euclidean ball in terms of factorials. For non-negative integer , these are:
The volume can also be expressed in terms of double factorials. For a positive odd integer , the double factorial is defined by
The volume of an odd-dimensional ball is
There are multiple conventions for double factorials of even integers. Under the convention in which the double factorial satisfies
the volume of an -dimensional ball is, regardless of whether is even or odd,
Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume:
Approximation for high dimensions
Stirling's approximation for the gamma function can be used to approximate the volume when the number of dimensions is high.
In particular, for any fixed value of the volume tends to a limiting value of 0 as goes to infinity. Which value of maximizes depends upon the value of ; for example, the volume is increasing for , achieves its maximum when , and is decreasing for .
Relation with surface area
Let denote the hypervolume of the -sphere of radius . The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by:
Thus, inherits formulas and recursion relationships from , such as
There are also formulas in terms of factorials and double factorials.
Proofs
There are many proofs of the above formulas.
The volume is proportional to the th power of the radius
An important step in several proofs about volumes of -balls, and a generally useful fact besides, is that the volume of the -ball of radius is proportional to :
The proportionality constant is the volume of the unit ball.
This is a special case of a general fact about volumes in -dimensional space: If
is a body (measurable set) in that space and is the body obtained by stretching in all directions by the factor then the volume of equals times the volume of . This is a direct consequence of the change of variables formula:
where and the substitution was made.
Another proof of the above relation, which avoids multi-dimensional integration, uses induction: The base case is , where the proportionality is obvious. For the inductive step, assume that proportionality is true in dimension . Note that the intersection of an n-ball with a hyperplane is an -ball. When the volume of the -ball is written as an integral of volumes of -balls:
it is possible by the inductive hypothesis to remove a factor of from the radius of the -ball to get:
Making the change of variables leads to:
which demonstrates the proportionality relation in dimension . By induction, the proportionality relation is true in all dimensions.