Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T16:08:52.255Z Has data issue: false hasContentIssue false

Convex-normal (Pairs of) Polytopes

Published online by Cambridge University Press:  20 November 2018

Christian Haase
Affiliation:
Mathematik, FU Berlin, 14195 Berlin, Germany. e-mail: haase@math.fu-berlin.de e-mail: janhofmann@math.fu-berlin.de
Jan Hofmann
Affiliation:
Mathematik, FU Berlin, 14195 Berlin, Germany. e-mail: haase@math.fu-berlin.de e-mail: janhofmann@math.fu-berlin.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 2012, Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence between $k$- and $(k+1)$-convex-normality (for $k\ge 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $\text{Q}$, where the normal fan of $P$ is a refinement of the normal fan of $\text{Q}$, if every edge ${{e}_{P}}$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) ${{e}_{\text{Q}}}$ of $\text{Q}$, then $(P+\text{Q})\cap {{\mathbb{Z}}^{d}}=(P\cap {{\mathbb{Z}}^{d}})+(\text{Q}\cap {{\mathbb{Z}}^{d}})$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

References

[BG99] Bruns, W. and Gubeladze, J., Normality and covering properties ofaffine semigroups. J. Reine Angew. Math. 510(1999), 161178. http://dx.doi.Org/10.1515/crll.1999.044 Google Scholar
[BGT97] Bruns, W., Gubeladze, J., and Trung, N. G., Normalpolytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485(1997), 123160. Google Scholar
[EW91] Ewald, G. and Wessels, U., On the ampleness of invertible sheaves in complete projective toric varieties. Results Math. 19(1991), no. 3-4, 275278. http://dx.doi.Org/10.1007/BF03323286 Google Scholar
[Gubl2] Gubeladze, J., Convex normality of rational polytopes with long edges. Adv. Math. 230(2012), no. 1,372-389. http://dx.doi.Org/10.1016/j.aim.2011.12.003 Google Scholar
[HHM07] Haase, C., Hibi, T., and Maclagan, D., Mini-workshop: Projective normality of smooth toric varieties, abstracts from the mini-workshop: held August 12–18, 2007. Oberwolfach Reports 4(2007), 2283-2319. Google Scholar
[KS03] Kantor, J.-M. and Sarkaria, K. S., On primitive subdivisions of an elementary tetrahedron. Pacific J. Math. 211(2013), 123155. http://dx.doi.Org/10.2140/pjm.2003.211.123 Google Scholar
[LTZ93] Liu, J. Y., Trotter, L. E., Jr., and Ziegler, G. M., On the height of the minimal Hilbert basis. Results Math. 23(1993), no. 3-4, 374376. http://dx.doi.Org/! 0.1007/BF03322309 Google Scholar