Quantum Jumps of Normal Polytopes

We introduce a partial order on the set of all normal polytopes in Rd . This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in Rd , ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in NPol(d), called quantum jumps. The existence of extremal objects inNPol(d) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in NPol(5) have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in NPol(4) and NPol(5), found by a combination of the developed theory, random generation, and extensive computer search.