similarity metric for two characteristics I am trying to develop a way to compare items. Each item has the same two properties associated with it, say, $p_{1}$ and $p_{2}$.
All I am given are these two measures for each item, each of which may be given as a range, say $x_{1} \le p_{1} \le x_{2}$ and $y_{1} \le p_{2} \le y_{2}$.
Do you think that something like a Euclidean distance for two items under comparison would make sense?
For two items, say, $a$ and $b$, that metric might be
$\sqrt((x_{1_a}-x_{1_b})^2 + (x_{2_a}-x_{2_b})^2 + (y_{1_a}-y_{1_b})^2 + (y_{2_a}-y_{2_b})^2)$
Thank you.

IIUC, you have for each given item $i$ a rectangle $r(i) = [x_{min}(i), x_{max}(i)]\times[y_{min}(i), y_{max}(i)]$ in the $(x,y)$-plane where the properties $(p_1(i), p_2(i))$ of this item $i$ could be found. So one could think of this rectangle as the support of a uniform rectangular probability density function (pdf). Then, finding the distance between two items could be translated into finding the distance between those two pdfs.
There are many notions of distance between pdfs, see e.g. here. One particular intuitive one would be the Wasserstein metric.
