Is this a conclusion group for a new fundamental geometry problem? [closed]

Question

Let σ(n) represent all possible values of the types of different lengths of segments connected to each other among n points in the definition,such as in the plane,σ(3)=(1,2,3),σ(3)min=1,in the straight line，σ(3)=(2,3).σ(n)infmin express that the minimum value that guarantees that the set of n points within the definition is infinite number of types. Within the following definition, n has the following undetermined conclusions about σ(n): Smooth closed curves in a plane (guaranteed for any): σ(4)min=2,σ(4)infmin=3,σ(6)min=6,σ(6)infmin=12 Smooth closed curves in space: σ(4)min=3,σ(4)infmin=4 Smooth enclosed surfaces in space: σ(6)min=2,σ(6)infmin=3,σ(7)min=3,σ(7)infmin=5,σ(8)min=5,σ(8)infmin=7