This is following in multivariate analysis book.
The coordinates $x^T=[x_1,x_2, \ldots, x_p]$ of the points a constant distance $c$ from $\bar x$ satisfy $(x-\bar x)^T S^{-1} (x-\bar x) = c^2 $
When $p=1$, $(x-\bar x)^T S^{-1} (x-\bar x) = (x_1-\bar x_1)^2/s_{11}$ is the squared distance from $x_1$ to $\bar x_1$ in standard deviation units.
When $p=2$, the equation defines a hyperellipsoid centerd at $\bar x$. It can be shown using integral calculus that the volume of this hyperellipsoid is related to $|S|$. In particular,
Volume of $\{x: (x-\bar x)^T S^{-1} (x-\bar x) ≤ c^2\} = k_p |S|^{1/2} c^p$
where the constant $k_p=s \pi^{p/2} \Gamma(p/2)$
I understood when $p=1$. But I don't understand the equation of volume when $p=2$.
I don't know why it changed from equality to inequality and the reason (or process) for the equation to be established. Please explain it so that I can understand.