A property of the bidimensional sphere

It is natural to ask for a reasonable constant k having the property that any open set of area greater than k on a bidimensional sphere of area 1 always contains the vertices of a regular tetrahedron. We shall prove that it is sufficient to take k=3/4. In fact we shall prove a more general result. The interested reader will not have any problem in establishing that 3/4 is the best constant with this property.