How to define sparseness of a vector? I would like to construct a measure to calculate the sparseness of a vector of length $k$. 
Let $X = [x_i]$ be a vector of length $k$ such that there exist an $x_i \neq 0$ . Assume $x_i \geq 0$ for all $i$.
One such measure I came across is defined as $$\frac{\sqrt{k} - \frac{\|X\|_1}{{\|X\|_2}}} {\sqrt{k} -1}\;,$$ where $\|X\|_1$ is $L_1$ norm and $\|X\|_2$ is $L_2$ norm.
Here, $\operatorname{Sparseness}(X) = 0$ whenever the vector is dense (all components are equal and non-zero) and $\operatorname{Sparseness}(X) = 1$ whenever the vector is sparse (only one component is non zero).
This post only explains the when $0$ and $1$ achieved by the above mentioned measure.
Is there any other function defining the sparseness of the vector.

You could of course generalize your current measure
\begin{align}
 S(X) = \frac{\frac{k^{(1/m)}}{k^{(1/n)}} -\frac{\|X\|_m}{\|X\|_n} } {\frac{k^{(1/m)}}{k^{(1/n)}}-1}
\end{align}
while preserving your properties you specified.
An interesting special case could be $m = 1, n \to \infty$, in which case the expression simplifies to
\begin{equation}
 S(X) = \frac{k-\frac{\|X\|_1}{\|X\|_c}}{k-1}
\end{equation}
where $c = \infty$, (for some reason, mathjax refused to render when I inserted $\infty$ directly in the fraction)
