I'm working on a project with a professor, and he mentioned it is well known that if we have some simple convex closed curve $\gamma$ in $\mathbb{R}^2$ with positive lower and upper bounds on curvature ($L$ and $M$ respectively), then we can find some constant $c>0$ such that if $x$ is some point interior to the curve and $r < \text{diam } \gamma$, then the area of $B(x; r) \cap D \ge c r^2$, where $B(x,r)$ is the disk centered at $x$ with radius $r$, and $D$ is the open region bounded by $\gamma$.
I tried using Fubini's Theorem to estimate $ m_2(B(x;r) \cap D) = \int_{B(x;r)\cap D} dA $ by integrating $\int_{A+\epsilon}^{B-\epsilon} m_1(E_x) \ dx$ for $\epsilon > 0$, where $m(E_x)$ is the 1-dimensional Lebesgue measure of $ E_x = \{ y \in \mathbb{R} \ : \ (x,y) \in B(x;r) \}$, and $A$ and $B$ are the leftmost and rightmost $x$-coordinates of any point in $B(x;r)$ respectively. But, I ended up with something like $m_2 (B(x;r) \cap D) \ge cr^2 - 2cr\epsilon$, but $c$ depended on $\epsilon$.
I feel quite lost on this. I don't really know where to start, and I don't really know how to handle these types of geometric arguments.