Following up on the last session, we again discuss Poincaré inequalities. Recall that given an open set $\Omega\subseteq \mathbb{R}^n$ and a non-negative weight $w$ the Poincaré constant is the smallest constant $C>0$ such that $\inf_{c\in\mathbb{R}} \|f-c\|_{L^2(\Omega,w dx)} \le C \|\nabla f\|_{L^2(\Omega, w dx)}$ for all $f\in L^2(\Omega, w dx)$ smooth. Clearly, if $\Omega$ consists of two (or more) connected components plugging in a piecewise constant function yields that the Poincaré constant is $+\infty$. More generally, domains with weak connectivity allow construction of similar functions and therefore have large Poincaré constants. We will discuss and prove a result attributed to Cheff Cheeger, which relates the Poincaré constant to an isoperimetric quantity known as the Cheeger constant. The result may be understood as a converse statement to the above observation and be causally summarized by 'for the Poincaré constant to be large, the domain must have necessarily disconnected geometry'.