This was posted (by me) and solved before here.

Omid Hatami wrote: A ((2-line))is the area between two parallel lines. length of ((2-line)) is distance of two parallel lines. We have a circle with radius 1 and we have covered the circle with some ((2-lines)). Prove sum of lengths of ((2-lines)) is at least 2. It is very old classical problem (in Russia, at least; I even think it was invented here). All we need is to consider a unit sphere. Then each (2-line) (extended to two parallel planes) conclude a part of the sphere of area $2\pi\cdot$(length of (2-line)). As all 2-lines should cover the whole sphere (which is of area $4\pi$), we obtain the result.

Moreover, it holds if we replace a circle with radius 1 to any set of wight at least 2 (i.e. such set that any strip, which contains it, has a wight at least 2). This fact belongs to Bang. I think Fedja may tell more about this.

Fedor Petrov wrote: I think Fedja may tell more about this. Just three remarks. Firstly, it is clear from the proof that, for the circle, the inequality remains true for multiple coverings, i.e., if you cover a circle of width $1$ by several strips so that each point belongs to at least $k$ strips, then the sum of widths of the strips is at least $k$. This is no longer true for other regions. For instance, one can cover the equilateral triangle of width $1$ by $5$ strips of width $\frac13$ so that each point is covered at least twice. This shows that Bang's theorem is subtler than the inequality for the circle. Secondly, the following question is still open (as far as I know). Suppose that some region $R$ is covered by strips $S_j$ of widths $w_j$. Let $W_j$ be the width of $R$ in the direction of $S_j$, i.e., the width of the narrowest strip with boundary lines parallel to those of $S_j$ containing $R$. Is it true that we must always have $\sum_j \frac{w_j}{W_j}\geqslant 1$? Keith Ball proved it for regions symmetric with respect to the origin, but even the case when $R$ is an equilateral triangle is still unsettled. Thirdly, Bang's and Ball's theorems remain true in higher dimensions.