parmenides51 wrote: Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ? Note for example that points $M_1(t-1,(t-1)^2-1)$ and $M_2(t+1,(t+1)^2-1)$ both are on the parabola $y=x^2-1$ Note also that the line $M_1M_2$ is tangent to the parabole $y=x^2$ at point $N(t,t^2)$ middle of $M_1M_2$ So $M_1M_2\subset U$ is between the two parabolas, as required, and has length $2\sqrt{4t^2+1}$ as great as we want.