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How can we define distance between, say, two of them? And if we the distance between two of them and the distance of another one to the first, do we know its distance tö rhe second? What ways are possible to induce a distance on them. The middle point seems a reasonable one but if the distance is non-zero they can still overlap. Is there a way to make their distance zero when they touch at an arbitrary point?

Let's simplify things. Instead of triangles consider circles in three dimensional space.

Deschele Schilder
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    For me is the distance of two triangles equal to the minimum of distances $dist(A,B),$ where $A$ is an arbitrary point on the first and $B$ on the second triangle. This makes the distances of two overlaping triangles (even if they have a single common point) equal to zero. – Viera Čerňanová Aug 09 '21 at 14:03
  • @user376343 If we choose the maximal distance, then we get the property that $dist(A,B) = 0 \iff A = B,$ which is one of the required properties of a metric – Doug M Aug 09 '21 at 14:07
  • @user376343 such a distance will not be a metric: if $A$ overlaps $B$ and $B$ overlaps $C$, but $A$ and $C$ have no overlap, then $dist(A, C) > dist(A, B) + dist(B, C) $, violating the triangle inequality. – mrp Aug 09 '21 at 14:11
  • It is true that the practical naive "distance" I have written about before, is not the distance in the proper sense, as pointed out by Doug M and mrp. – Viera Čerňanová Aug 09 '21 at 15:35
  • Try to define for a "simpler" case, distance between circles. Also important which space those objects are embedded in (assume euclidean plane or $\mathbb{R}^2$). – z100 Aug 09 '21 at 16:19

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