How is the distance defined then?

R8kt wrote: How is the distance defined then? I think it is simply defined as the smallest distance of $O$ from a point of the segment. Under this assumption, let us first work with a single segment $AB$. If the projection $C$ of $O$ on the line $AB$ lies inside the segment $AB$, then we can split $AB$ in the two segments $AC$ and $BC$ leaving unvaried the sum, so wlog $C$ is outside the segment (wlog from the side of $B$, but it may also coincide with $B$). Now, let $\angle BOA=\theta$ and $\angle COB=\alpha$. We have that (if $a=AB$ and $h=d(O,AB)=OB$) $a=AC-BC=OC(\tan (\alpha+\theta)-\tan\alpha)=h\cos\alpha(tan(\theta+\alpha)-\tan\alpha))$. It is easy to check (even graphically) that $f(\alpha)=\cos\alpha(\tan(\theta+\alpha)-\tan\alpha)$ is increasing on $[0,\pi/2-\theta)$, and so it has its minimum in $0$. Therefore $a\geq h\cos0\tan\theta\geq h\theta$. Therefore, it $\theta_i$ is the angle formed by the $i$-th segment, we have $\sum_{i=1}^n \frac{a_i}{h_i}\geq \sum_{i=1}^n \theta_i\geq 2\pi$ as wanted.