Extremal problems for convex lattice polytopes: a survey
نویسندگان
چکیده
What is the minimal area that a convex lattice polygon with exactly n vertices can have? This question, and several similar ones, are considered in this survey.
منابع مشابه
Geometry of random (-1/+1)-polytopes A survey of recent results on random {−1, 1}-polytopes in Asymptotic Convex Analysis
Random {−1, 1}-polytopes demonstrate extremal behavior with respect to many geometric characteristics.
متن کاملExtremal Edge Polytopes
The edge polytope of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k ∈ {2, 3, 5}, we determine the maximal number of vertices of kneighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.
متن کاملConvex Polytopes: Extremal Constructions and f -Vector Shapes
These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f -vectors, and extremal constructions. The study of f -vectors has had huge successes in the last forty years. The most fundamental one is undoubtedly the “g-theorem,” conjectured by McMullen in 1971 and proved by Billera & Lee and Stanley in 1980, which characte...
متن کامل7 Lattice Points and Lattice Polytopes
Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation theory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the theory of lattice polytopes and only sketch their numerous applicatio...
متن کاملConvex Lattice Polytopes and Cones with Few Lattice Points Inside, from a Birational Geometry Viewpoint
It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P ⊂ R of positive finite volume, such that P ⋂ Z consists of the point 0 and vertices of P (cf., e.g. [10], [36]). Likewise, Q−factorial terminal toric singularities essentially correspond to lattice simplexes with no lattice points inside or on the boundary (excep...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007