Let $0<m\le n$, let $x_0\in\mathbb R^n$, and let $R>0$. Let either $\Sigma^m\subset B(x_0,R)\subset\mathbb R^n$ be a smooth properly embedded minimal submanifold without boundary in $B(x_0,R)$, with associated mass measure $\mu=\mathcal H^m\big|_\Sigma$, or more generally let $V$ be a stationary integral $m$-varifold in $B(x_0,R)$, with associated weight measure $\mu=\|V\|$. Define the density-ratio function $\Theta:(0,R)\to[0,\infty)$ by

\begin{align*} \Theta(\rho):=\Theta(\mu,x_0,\rho):=\frac{\mu(B(x_0,\rho))}{\theta_m\rho^m}, \end{align*}

where $\theta_m:=\mathcal L^m(B(0,1))$. Then $\Theta$ is nondecreasing. More precisely, for $0<\sigma<\rho<R$,

\begin{align*} \Theta(\mu,x_0,\rho)-\Theta(\mu,x_0,\sigma) =\frac{1}{\theta_m}\int_{B(x_0,\rho)\setminus B(x_0,\sigma)} \frac{|(x-x_0)^\perp|^2}{|x-x_0|^{m+2}}\,d\mu(x), \end{align*}

where $(x-x_0)^\perp$ denotes the component of $x-x_0$ normal to the approximate tangent plane of $V$ at $\mu$-almost every $x$, and in the smooth submanifold case this is the [orthogonal projection](/theorems/437) onto $(T_x\Sigma)^\perp$.