Busy-body curves have to be long

Question

Question:

Prove that for every positive constant $C$ there exists an $\epsilon > 0$ such that any curve in $\mathbb{R}^{2}$ whose image has points at distance $\leq \epsilon$ from each point of the unit square $[0,1]^{2}$ must have length $\geq C$.

My problem:

I have difficulty in understanding the question

• how can a point in the image of a curve have a distance $\leq \epsilon$ from all the points of the unit solid square? I assume that $\epsilon$ can be chosen as a small value.
• Also, suppose the length of one curve is $C^{\prime}$, and we can find an $\epsilon$, then this $\epsilon$ holds for all $C > C^{\prime}$ for this particular curve. How could this curve have length $\geq C$? (this is also related to understanding "any" curve, that is, for any curve, we have a $C$, or for any $C$ we have all curves satisfying blablabla).

Any explanation to help me understand this problem? Thank you!

Edit:

Thank you @Robert Israel for making my first bullet clear. However, I am still puzzling about my second bullet. I re-state my second bullet as follows,

Prove that, given any curve in $\mathbb{R}^{2}$, for every positive constant $C$ there exists an $\epsilon > 0$ such that for every point from the unit square $[0,1]^{2}$ we can always find at least one point from the image of this curve at distance $\leq \epsilon$, then this curve must have length $\geq C$.

Does my statement have the same meaning as the problem does? If so, what does the $C$ do in the statement? It seems only the $\epsilon$ is related to the curve.

Regarding the example below. The curve has a length of 4.75, and we find an $\epsilon = 1/8$ to satisfy all the requirements, but since $C$ is irrelated to the $\epsilon$, I can claim the statement holds for arbitrary $C >0$, so the length of this curve should be arbitrarily large. A contradiction. Now I was wondering if there must be some places where I misunderstood the statement of the problem.

Answer 1

• For example, the red curve has distance $\le 1/8$ from all points of the unit square.

• Suppose for example you were to claim that $\epsilon = 1/8$ works for $C = 2$. Since my red curve works for $\epsilon = 1/8$, it would imply that my red curve has length $\ge 2$. Well, my red curve has length $4.75$, and that's greater than $2$, so no problem. If you were to find some other curve with length $< 2$ that is within distance $1/8$ of every point of the square, that would contradict your claim. But if there is no such curve with length $< 2$, your claim is true.