An Upper Bound Theorem concerning lattice polytopes
نویسندگان
چکیده
R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings. We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the delta-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.
منابع مشابه
CAYLEY DECOMPOSITIONS OF LATTICE POLYTOPES AND UPPER BOUNDS FOR h-POLYNOMIALS
We give an effective upper bound on the h-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley su...
متن کاملar X iv : 0 70 6 . 41 78 v 1 [ m at h . C O ] 2 8 Ju n 20 07 Lattice polytopes of degree 2 Jaron Treutlein
Abstract. A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i > 0 interior lattice points. We give a new proof for this theorem and classify polygons with maximal volume. Then we show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimension. From a theorem of Victor Batyrev the finiteness of ...
متن کاملar X iv : 0 70 6 . 41 78 v 2 [ m at h . C O ] 2 1 Fe b 20 08 Lattice Polytopes of Degree 2
Abstract. A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i > 0 interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. The finiteness of lattice polytopes of degree 2 up to standard pyramids and affine unimodular transformation follows from a theore...
متن کاملLattice Polytopes of Degree
Abstract. A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i > 0 interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. The finiteness of lattice polytopes of degree 2 up to standard pyramids and affine unimodular transformation follows from a theore...
متن کاملNotes on lattice points of zonotopes and lattice-face polytopes
Minkowski’s second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski’s bound by replacing the volume by the lattice point enumerator of a convex body. To this we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our resu...
متن کامل