the paper:

\[thm1\] Let $G(\chi_n,r_n)$ be a random geometric graph model on the d-dimensional torus with respect to some norm $\| \cdot \|$ and a threshold parameter that satisfies $n^{ (\delta^*-2)/d } \leq r_n \leq n^{- \delta^*/d}$ for some fixed $ \delta^*>0$. Let $\tau_n = \nu (r_n/2)^d$, where $\nu$ is the volume of the unit ball under the norm $\|\cdot \|$. (That is, $\tau_n$ is the volume of the ball of diameter $r_n$.) Fix $\delta >0$ and $\varepsilon > 0$. Let $F_n$ be the event that the following happen: (a) There is a ball $A$ of diameter $r_n$ such that any convex set $S\subseteq A$ with $\lambda (S) >
(\varepsilon/16) \tau_n$ (where $\lambda$ denotes Lebesgue measure) satisfies $$\left|\frac{|\chi_n(S)|}{\sqrt{2 \delta \mu_n}} - \frac{\lambda(S)}{\tau_n}\right| < \varepsilon\, ,$$ and (b) for any convex set $S\subseteq A^c$ with $\lambda(S) > (\varepsilon/16)\tau_n$, such that $S$ lies in some ball of diameter $r_n$, $$\frac{|\chi_n(S)|}{\sqrt{2 \delta \mu_n}} < \varepsilon\,\frac{\lambda(S)}{\tau_n}\, .$$ Then the conditional probability of the event $F_n$ given that $|E| \geq (1+\delta) \mu_n$ tends to $1$ as $n\to \infty$.

The convexity requirement is probably not optimal, but forces the set $S$ to be sufficiently “nice” to preclude sets which are either sparse but of large measure 