Define diameter of a set S which is the subset of a Euclidean space?

Answer 1

If S is a set of points in a metric space X, the diameter of S is the longest distance between two points in S.

Assuming you are only interested in two- or three-dimensional Euclidean spaces, the distance calculation is straightforward.

For a higher-dimension Euclidean space, efficient calculation of a subset's diameter is more complex.

Answer 2

The diameter equation follows Violet. According to Munkers Topology, daimA = sup{d(a1,a2): a1, a2 are in A} for any metric d on the metric space X. This means that it does not have to be euclidean and you just have to compute the distances according to the metric and find the sup.

Answer 3

The diameter is the suprememum of the distance between any two points of S.

D(S) = sup { d(x,y) | x,y in S }

For instance, if S is the open sphere of radius R around the origin, the distances between any two points x,y in S is
d(x,y) < 2R
Moreover we can approach 2R as close as we want by picking points close enough to the boundary of the sphere. Therefore,
sup d(x,y) = 2R
so that
D(S) = 2R.