Let's start with two concentric circles of radii $r<R$. Then we put two sticks inside the outer circle while avoiding the inner circle, say $AB$ and $CD$.
Then we compare the length of inner part $AP+DP$ with the exterior part $BP+CP$. It seem that there exist a lower bound for the difference: $(AP+DP)-(BP+CP)\ge\epsilon$ for all such sticks with a uniform constant $\epsilon>0$.
I don't if this is true. (For specific reason we can use $r=1$ and $R=2$).
If it is, could you give any hint how to prove it? Thanks!
Edit: Now I know there is no uniform lower bound since the distance goes to zero if
the point $C$ moves close to $A$;
the lengths of both sticks goes to zero: $AB\to0$ and $CD\to0$.
The second case can be viewed as a special case of the first one. So a reasonable modification of my question is:
- Will there exist $\epsilon=\epsilon(\delta)>0$ such that for if the distance of the two ends $AC\ge\delta$, then the difference $(AP+DP)-(BP+CP)\ge\epsilon$?