MathDB

A regular tetrahedron with unit edge length is given. Prove that:
(a) There exist four points on the surface

S

of the tetrahedron, such that the distance from any point of the surface to one of these four points does not exceed

1/2

;
(b) There do not exist three points with this property.
The distance between two points on surface

S

is defined as the length of the shortest polygonal line going over

S

and connecting the two points.

Problem Statement