Gorenstein simplices and the associated finite abelian groups

It is known that a lattice simplex of dimension $d$ corresponds a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension $d$. In this paper, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals $p,p^2$ and $pq$, where $p$ and $q$ are prime numbers with $p \neq q$. Moreover, we compute the volume of the dual simplices of Gorenstein simplices.Comment: 18 pages, to appear in European Journal of Combinatoric

Cayley sums and Minkowski sums of 22-convex-normal lattice polytopes

In this paper, we discuss the integer decomposition property for Cayley sums and Minkowski sums of lattice polytopes. In fact, we characterize when Cayley sums have the integer decomposition property in terms of Minkowski sums. Moreover, by using this characterization, we consider when Cayley sums and Minkowski sums of $2$-convex-normal lattice polytopes have the integer decomposition property. Finally, we also discuss the level property for Minkowski sums and Cayley sums.Comment: 10 page

Reflexive polytopes arising from perfect graphs

Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In the present paper, by virtue of the algebraic technique on Gr\"onbner bases, a new class of reflexive polytopes which possess the integer decomposition property and which arise from perfect graphs will be presented. Furthermore, the Ehrhart $\delta$-polynomials of these polytopes will be studied.Comment: 13 page

Optimal size of central government and agglomeration

Though the central government uses neither a transfer nor a regional allocation policy, it can affect the distribution of the population. This paper analyzes the optimal government policy and examines whether or not the government should take into account agglomeration without a regional redistribution policy. The optimal size of central government depends on the degree of increasing returns in the private and the public sector. When the central government shows a much lower degree of increasing returns in contrast to the private sector, it should decrease the provision of the public good. As a result, the central government limits agglomeration. If the central government does not consider its effect on agglomeration, it is too large in size, and it causes too much agglomeration.Agglomeration, Central government, Regional distribution

Reflexive polytopes arising from partially ordered sets and perfect graphs

Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gr\"obner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furthermore, the result will give a polyhedral characterization of perfect graphs. Finally, we will investigate the Ehrhart $\delta$-polynomials of these reflexive polytopes.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1703.0441