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Suppose we have a space like $\partial([0,1]^2)$, the boundary of the unit box, simple a square. I'm looking for a metric that would measure the distances that must be traversed to get from one point to another, given that one cannot leave the set $\partial([0,1]^2)$.

Let $\rho$ be that metric. It should be something like.

$$\rho(a,x) = \min \left\{ a_1 + x_2 + x_1, 1-a_1 + x_2 + 1-x_1 \right\}$$

for $a_2 \neq x_2$.

If $a_2 = x_2$, $$\rho(a,x) = \vert a_1 - x_2 \vert. $$

Is there a name for this kind of thing? It seems simple enough that it might show up somewhere. I'm tempted by the perimeter metric, as we can only travel on the perimeter of the unit box.

user178543
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1 Answers1

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I would consider this as the intrinsic metric of $\partial([0,1]^2)$.

Another point of view is that this metric is nothing more than the standard metric on $[0,4]/\sim$, where $0 \sim 4$, which can also be seen as a metric on the unit circle (but you need an appropriate "rescaling").

Silvia Ghinassi
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