Volume of a hypersphere in the 1-norm


$\lVert\x\rVert \leq R ,$

where $\lVert\x\rVert$ denotes the usual Euclidean norm (also known as the 2-norm),

$\lVert\x\rVert := \sqrt{x_1^2+\dotsb+x_n^2} .$

Today, I’d like to consider the problem of computing the volume of an $n$-dimensional hyphersphere in the 1-norm (also known as the Manhattan distance or taxicab norm), which is defined by

$\lVert\x\rVert_1 := \lvert x_1\rvert+\dotsb+\lvert x_n\rvert .$

The volume of the 1-norm hypersphere is given by the expression

$V_n(R) := \frac{(2R)^n}{n!} ,$

as we will show by induction on $n$. In the base case $n=1$ one has

$\newcommand{\d}{\,\mathrm{d}} V_1(R) = \int_{-R}^R\d x_1 = 2R ,$

as required. Now suppose that the formula holds in dimension $n-1$. Then we have

\begin{align*}
V_n(R) &= \int\limits_{\lvert x_1\rvert\leq R}\;\int\limits_{\lvert x_1\rvert+\lvert x_2\rvert\leq R}\dotsi\int\limits_{\lvert x_1\rvert+\dotsb+\lvert x_n\rvert\leq R}\d x_n\dotsm\d x_1 \\
&= \int\limits_{\lvert x_1\rvert\leq R}\;\int\limits_{\lvert x_2\rvert\leq R-\lvert x_1\rvert}\dotsi\int\limits_{\lvert x_2\rvert\dotsb+\lvert x_n\rvert\leq R-\lvert x_1\rvert}\d x_n\dotsm\d x_1 \\
&= \int\limits_{\lvert x_1\rvert\leq R} V_{n-1}\bigl(R-\lvert x_1\rvert\bigr) \d x_1 \\
&= \int_{-R}^R \frac{2^{n-1}(R-\lvert x_1\rvert)^{n-1}}{(n-1)!} \d x_1 \\
&= 2\int_{0}^R \frac{2^{n-1}(R-x_1)^{n-1}}{(n-1)!} \d x_1 \\
&= \frac{2^n}{(n-1)!}\biggl[-\frac{1}{n}(R-x_1)^n\biggr]_0^R \\
&= \frac{(2R)^n}{n!}
\end{align*}

By induction, the formula holds for all positive integers $n$.