I made up this question.
A police officer's job is to patrol the perimeter of an elliptical lake. They have a constant (unknown) swimming speed and a constant (unknown) running speed. Their quickest path between any two points on the perimeter never involves both swimming and running. What is the maximum value of $e$, the eccentricity of the ellipse?
Proof that $e$ has an upper bound
If $e\approx 1$ then the lake looks almost like a straight river. Consider a point on one side, and another point on the other side further "downstream". Clearly, the quickest path between the points could involve swimming and running.
We can prove that if $e=0$ (circle) then the quickest path never involves swimming and running. Suppose $e=0$ and the quickest path between two points involves swimming and running, i.e. the path includes a chord and arc that share an endpoint. Let $\alpha=$ angle subtended by the chord ($0<\alpha<\pi$), and $\theta=$ angle subtended by the chord-arc pair.
Assuming the radius of the circle is $1$, the time for this path is
$T=\dfrac{2\sin (\alpha/2)}{v_{\text{swim}}}+\dfrac{\theta-\alpha}{v_{\text{run}}}$
$\dfrac{d^2 T}{d\alpha^2}=-\dfrac{\sin (\alpha/2)}{2v_{\text{swim}}}<0$
So the minimum value of $T$ occurs at one of the endpoints, i.e. $\alpha=0$ or $\alpha=\theta$, which implies that the path involves either only running or only swimming, contradiction. Therefore if $e=0$ then the quickest path never involves swimming and running.
So $e$ must have an upper bound.