A circle in Euclidean Geometry are all the point at same fixed distance (radius) from a single point called the center.
That fixed distance is only from every point to the center, but there are (obviously) different distances between point to point, by definition they are different points because they have a non-zero distance between them.
Well I wonder, (I've just imagine it and don't know if exist), an object of a special shape, of certain dimension, in a certain metric.. to have a single non zero distance between all points?
I mean, a shape/or metric, in wich points although perhaps they may have different distances between them, (as any known object in Euclides metric the circle, a plane, a sphere, a line, etc.), the special feature (or the special metric feature) would be that measuring those points from other metric, then all points are the same "distance" each other
thanks
Best Answer
Define a metric $d$ such that $d(x,y)=1$ if $x \not= y$ and $d(x,y)=0$ if $x=y.$ We should also check that $d$ satisfies the metric conditions:
i) $d(x,y) \geq 0$ for all $x$ and $y.$
ii) $d(x,y)=0$ iff $x=y.$ This is true by definition.
iii) $d(x,y)=d(y,x)$ which is obvious.
iv) $d(x,z) \leq d(x,y)+d(y,z)$ holds again by definition.
Therefore $d$ is a metric.