ABSTRACT:

1) Introduction:

I explain the difference between an "absolute" geometric optimization

problem and a "relative" geometric optimization problem. To see this

difference I give a basic description of the isoperimetric problem both

on the plane and on the sphere (without giving proofs). I show some

applications of "relative" geometric problems in real life, and try to

give some answers to the best procedure of dividing a set into two

subsets in such a way that some geometric measure is "optimized".

2) Recent results:

I give the proof of a new "relative" isodiametric inequality, first in

the general case (for convex surfaces) then in the more restricted case

(adding the assumption of central symmetry) and finally in the

particular interesting example of the cube, and show how the bounds

sharpen in each step. The proof in the case of the cube is done in a

very geometric way, easy to visualize by "coloring".