Given a collection $V$ of $N$ points in a metric space $(M, d)$, I define a circuit as a sequence $w = (w_1, w_2, …, w_N)$ which visits each point in $V$ and has a length given by

$$|w| = sum_{i=1}^N d(w_i, w_{i+1})$$

where $w_{N+1} = w_1$. I am interested in

$$omega_{text{min}} = min_w |w|$$

$$omega_{text{max}} = max_w |w|$$

and how they may be related to the constants

$$C_p = sum_{v in V} d(v, p)$$

As such, I considered the smallest such constant

$$C = min_{p in M} C_p$$

and it seemed that

$$C le omega_{text{max}} le 2C$$

for all collections $V$. *Have I missed any obvious counter examples?*

So far I have only found a proof for $omega_{text{max}} le 2C$. For any circuit $w$, consider the sum of the perimeters of all triangles with vertices $w_i$, $w_{i+1}$ and $p$. This should evaluate to $|w| + 2C_p$ and by the triangle inequality this is at least $2|w|$.

Intuition tells me that $omega_{text{max}} = 2C$ iff the points in $V$ are colinear, since this is when we have equality in the triangle inequality. *Is this correct?*

Finally, I know examples where $omega_{text{min}} le C$ and other examples where $C le omega_{text{min}}$ but for large $N$ it seems that the former is more likely. *Is there a condition on $V$ which requires $omega_{text{min}} le C$ regardless of the metric space?*

Any help towards answering these questions would be greatly appreciated.