Let $T$ be a set of $n\ge 3$ points in the plane such that not all of them lie in a common line. Pick two distinct points $\{a=\left( \begin{array}{c} a_{1} \\a_{2} \end{array} \right) ,b=\left( \begin{array}{c} b_{1} \\b_{2} \end{array} \right)\}$ uniformly at random from $T$. Let $R$ be the rectangle whose sides are horizontal and vertical, and whose two diagonally opposite vertices are $\{a,b\}$. Let $X$ be the length of horizontal side of $R$ and $Y$ be the length of its vertical side (i.e. $X=|a_{1}-b_{1}|$ and $Y=|a_{2}-b_{2}|$).

Assuming $\mathbb{E}(X)=\mathbb{E}(Y)$, how small can be expected value of random variable $Z := \min(X,Y)$? What lower bounds can one get for $\mathbb{E}(Z)$?