# mg.metric geometry – Bounds on the lengths of circuits in a metric space

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.