submitted at the Oberwolfach Conference “Combinatorial Convexity and Algebraic Geometry” 26.10–01.11, 1997 Throughout, we fix the notation M := Z and MR := R . Given convex lattice polytopes P, P ′ ⊂ MR, we have (M ∩ P ) + (M ∩ P ) ⊂ M ∩ (P + P ), where P + P ′ is the Minkowski sum of P and P , while the left hand side means {m+m | m ∈ M ∩ P,m ∈ M ∩ P }. Problem 1 For convex lattice polytopes P...

In this paper we have shall generalize Shearer’s entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.

We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in R. In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m1,m2, . . . ,mk facets respectively is bounded from above by

‎We consider four-dimensional lie groups equipped with‎ ‎left-invariant Lorentzian Einstein metrics‎, ‎and determine the harmonicity properties ‎of vector fields on these spaces‎. ‎In some cases‎, ‎all these vector fields are critical points for the energy functional ‎restricted to vector fields‎. ‎We also classify vector fields defining harmonic maps‎, ‎and calculate explicitly the energy of t...

Given a convex body K ⊂ Rn with the barycenter at the origin we consider the corresponding Kähler-Einstein equation e = detDΦ. If K is a simplex, then the Ricci tensor of the Hessian metric DΦ is constant and equals n−1 4(n+1) . We conjecture that the Ricci tensor of D Φ for arbitrary K is uniformly bounded by n−1 4(n+1) and verify this conjecture in the two-dimensional case. The general case r...

We derive tight bounds for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P1 ⊕ P2, of two ddimensional convex polytopes P1 and P2, as a function of the number of vertices of the polytopes. For even dimensions d ≥ 2, the maximum values are attained when P1 and P2 are cyclic d-polytopes with disjoint vertex sets. For odd dimensions d ≥ 3, the maximum values are attained when ...

Let P and Q be finite sets of points in the plane. In this note we consider the largest cardinality of a subset of the Minkowski sum S ⊆ P ⊕ Q which consist of convexly independent points. We show that, if |P | = m and |Q| = n then |S| = O(m2/3n2/3 + m + n).

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let δ(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine δ(d, k) for small instances. We show that δ(3, 4) = 7 and δ(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby δ(d, ...

We derive semiclassical quantization conditions for particles with spin. These generalize the Einstein-Brillouin-Keller quantization in such a way that, in addition to the Maslov correction, there appears another term which is a remnant of a non-Abelian geometric or Berry phase. This correction is interpreted in terms of a rotation angle for a classical spin vector.