Suppose we've a convex quadrilateral both of whose diagonals have length 2. Is it true that the product of the lengths of the quadrilateral's sides must be less than or equal to 5?
If we require, in addition, that the diagonals of the quadrilateral are perpendicular, this becomes equivalent to the inequality of this question.
A product of 5 is achieved when the diagonals are perpendicular and 3 of the vertices are colinear. I am also curious whether a similar configuration is optimal in the case where the diagonals have different lengths x and y. If so, we would obtain a bound on the product in terms of x and y by computing the product for this configuration.