Deriving The Volume of An N-ball - Balls in Lp Norms

Balls in Lp Norms

There are also explicit expressions for the volumes of balls in Lp norms. The Lp norm of the vector x = (x₁, ..., x_n) in Rn is, and an Lp ball is the set of all vectors whose Lp norm is less than or equal to a fixed number called the radius of the ball. The case p = 2 is the standard Euclidean distance function, but other values of p occur in diverse contexts such as information theory, coding theory, and dimensional regularization.

The volume of an Lp ball of radius R is:

These volumes satisfy a recurrence relation similar to the one dimension recurrence for p = 2:

Notice that for p = 2, we recover the recurrence for the volume of a Euclidean ball because .

For example, in the cases p = 1 and p = ∞, the volumes are:

These agree with elementary calculations of the volumes of cross-polytopes and hypercubes.

For most values of p, the surface area of an Lp sphere (the boundary of an Lp ball) cannot be calculated by differentiating the volume of an Lp ball with respect to its radius. While the volume can be expressed as an integral over the surface areas using the coarea formula, the coarea formula contains a correction factor that accounts for how the p-norm varies from point to point. For p = 2 and p = ∞, this factor is one. However, if p = 1, then the correction factor is : The surface area of an L1-(n − 1)-sphere of radius R is times the derivative at R of the volume of an L1-n-ball. For most values of p, the constant is a complicated integral.

The volume formula can be generalized even further. For positive real numbers p₁, ..., p_n, define the unit (p₁, ..., p_n) ball to be:

The volume of this ball is: