Given a quadrilateral with side lengths $a,b,c,d$ and diagonal lengths $e,f$, let us use the shortcut $s=\min\{a,b,c,d\}$ (”smallest side”) and $\delta=\max\{e,f\}$ (”largest diagonal”).
My questions are:
- What is the infimum $I_1$ of $\delta/s$, taken over the set $M$ of all quadrilaterals whose diagonals intersect.
- What is the infimum $I_2$ of $\delta/s$, taken over all quadrilaterals from $M$ with the additional property $\min\{e,f\}\le s$.
By definition $I_2\ge I_1$ (the infimum is taken over a smaller set for $I_2$), and also by definition $$I_1\ge1\iff\text{$\delta\ge s$ for every quadrilateral from $M$}\iff I_2\ge1\text.\quad(*)$$
I can prove that the second assertions of $(*)$ is actually true, using only the triangle inequality with the intersection point of the diagonals as intermediate point for the sides (that is, after an appropriate reformulation of the question, this estimate holds in every metric space).
However, actually I am interested throughout only in the Euclidean plane.
In this case, the set $M$ is the set of convex quadrilaterals.
I conjecture that in the Euclidean plane we do not have $I_1=I_2=1$.
More precisely, the best possible upper bounds I was able to prove so far are $I_1\le\sqrt2$ (considering the square) and $I_2\le\sqrt3$ (considering the particular rhombus $a=b=c=d=e$), and I conjecture that equality holds for $I_2$; perhaps also equality holds for $I_1$.
Indeed, at least under local perturbations the quadrilateral providing the upper estimate is the one with the smallest quotient $\delta/s$. (Of course, this does not prove that it is a global minimum.) Moreover, I can prove $\delta/s\ge\sqrt2$ for some classes of convex quadrilaterals (for example, if the intersection point is the middle point of both diagonals), though so far not for the general case. (My heuristical argument for the conjecture $I_2\ge\sqrt3$ is too lengthy with too many gaps to present here.)
Background of the question was a problem from entertainment mathematics which was essentially a reformulation of the second assertion in $(*)$ in the Euclidean plane. Since as mentioned this assertion holds in every metric space and not only in the Euclidean plane, it is probably natural to ask whether actually more can be said in the Euclidean plane.