I came up with what seems quite a beautiful geometric construction solving the problem described below. On a compact convex set consider the function
where we denote In other words, “hanging” in the point we measure the “width” of the set along the first axis of the Cartesian system (note that actually doesn’t depend on ). In the same fashion we define functions ( measures the “width” along the -th axis). Now consider the following functionals of :
Namely, is the maximum sum of “widths”, whereas is the sum of maximum “widths”, hence Moreover, for some archetypal examples of convex polyhedra: hyperrectangle (box) and rectangular simplex, and it is easy to begin thinking that we have equality for any convex polyhedra. To give a partial justification for this intuition (which is though incorrect in general), consider rectangular convex polyhedra: to build such a polyhedron in we fix a pair of points on each coordinate axis and then take the convex hull of these points. We can easily prove the following geometric fact about such polyhedra:
For any rectangular (in the sense defined above) convex polyhedron in that is, rectangular convex quadrilateral, it holds
However already in one may find a convex polyhedron (in fact a rectangular one), such that
Point out a sequence , , of [‘shear’] convex polyhedra, for which
Later on I will describe the solution to this problem.