  A matrix is called totally positive if every minor of it is positive. Such
matrices are well studied and have numerous applications in Mathematics and
Computer Science. We study how many times the value of a minor can repeat in a
totally positive matrix and show interesting connections with incidence
problems in combinatorial geometry. We prove that the maximum possible number
of repeated $d \times d$-minors in a $d \times n$ totally-positive matrix is
$O(n^{d-\frac{d}{d+1}})$. For the case $d=2$ we also show that our bound is
optimal. We consider some special families of totally postive matrices to show
non-trivial lower bounds on the number of repeated minors. In doing so, we
arrive at a new interesting problem: How many unit-area and axis-parallel
rectangles can be spanned by two points in a set of $n$ points in the plane?
This problem seems to be interesting in its own right especially since it seem
to have a flavor of additive combinatorics and relate to interesting incidence
problems where considering only the topology of the curves involved is not
enough. We prove an upper bound of $O(n^{\frac{4}{3}})$ and provide a lower
bound of $n^{1+\frac{1}{O(\log\log n)}}$.
