Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0004
Gabriele Balletti, Christopher Borger
{"title":"Families of 3-dimensional polytopes of mixed degree one","authors":"Gabriele Balletti, Christopher Borger","doi":"10.1142/9789811200489_0004","DOIUrl":"https://doi.org/10.1142/9789811200489_0004","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125359384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0002
Katharina Jochemko
These notes are based on a five-lecture summer school course given by the author at the “Summer Workshop on Lattice Polytopes” at Osaka University in 2018. We give a short introduction to the theory of valuations on lattice polytopes. Valuations are a classical topic in convex geometry. The volume plays an important role in many structural results, such as Hadwiger’s famous characterization of continuous, rigid-motion invariant valuations on convex bodies. Valuations whose domain is restricted to lattice polytopes are less well-studied. The Betke-Kneser Theorem establishes a fascinating discrete analog of Hadwiger’s Theorem for lattice-invariant valuations on lattice polytopes in which the number of lattice points — the discrete volume — plays a fundamental role. From there, we explore striking parallels, analogies and also differences between the world of valuations on convex bodies and those on lattice polytopes with a focus on positivity questions and links to Ehrhart theory.
{"title":"A brief introduction to valuations on lattice polytopes","authors":"Katharina Jochemko","doi":"10.1142/9789811200489_0002","DOIUrl":"https://doi.org/10.1142/9789811200489_0002","url":null,"abstract":"These notes are based on a five-lecture summer school course given by the author at the “Summer Workshop on Lattice Polytopes” at Osaka University in 2018. We give a short introduction to the theory of valuations on lattice polytopes. Valuations are a classical topic in convex geometry. The volume plays an important role in many structural results, such as Hadwiger’s famous characterization of continuous, rigid-motion invariant valuations on convex bodies. Valuations whose domain is restricted to lattice polytopes are less well-studied. The Betke-Kneser Theorem establishes a fascinating discrete analog of Hadwiger’s Theorem for lattice-invariant valuations on lattice polytopes in which the number of lattice points — the discrete volume — plays a fundamental role. From there, we explore striking parallels, analogies and also differences between the world of valuations on convex bodies and those on lattice polytopes with a focus on positivity questions and links to Ehrhart theory.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131920892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0016
Yonggyu Lee
{"title":"A short survey on Tesler matrices and Tesler polytopes","authors":"Yonggyu Lee","doi":"10.1142/9789811200489_0016","DOIUrl":"https://doi.org/10.1142/9789811200489_0016","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"139 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115253588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0025
Johanna K. Steinmeyer
{"title":"On the faces of simple polytopes","authors":"Johanna K. Steinmeyer","doi":"10.1142/9789811200489_0025","DOIUrl":"https://doi.org/10.1142/9789811200489_0025","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121562213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0013
Magda L Hlavacek
{"title":"Cubical Dehn–Sommerville equations and self-reciprocal cubical complexes","authors":"Magda L Hlavacek","doi":"10.1142/9789811200489_0013","DOIUrl":"https://doi.org/10.1142/9789811200489_0013","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133037185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0014
Óscar Iglesias-Valiño
{"title":"Hollow lattice polytopes: Latest advances in classification and relations with the width","authors":"Óscar Iglesias-Valiño","doi":"10.1142/9789811200489_0014","DOIUrl":"https://doi.org/10.1142/9789811200489_0014","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123515731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0005
Mónica Blanco
{"title":"Some lattice parallelepipeds with unimodular covers","authors":"Mónica Blanco","doi":"10.1142/9789811200489_0005","DOIUrl":"https://doi.org/10.1142/9789811200489_0005","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130431654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0007
F. Castillo
This is a hands on introduction to McMullen’s Polytope Algebra. More than interesting on its own, this algebra was McMullen’s tool to give a combinatorial proof of the g-theorem.
{"title":"A pithy look at the polytope algebra","authors":"F. Castillo","doi":"10.1142/9789811200489_0007","DOIUrl":"https://doi.org/10.1142/9789811200489_0007","url":null,"abstract":"This is a hands on introduction to McMullen’s Polytope Algebra. More than interesting on its own, this algebra was McMullen’s tool to give a combinatorial proof of the g-theorem.","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129358451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.1142/9789811200489_0001
Johannes Hofscheier
{"title":"Introduction to toric geometry with a view towards lattice polytopes","authors":"Johannes Hofscheier","doi":"10.1142/9789811200489_0001","DOIUrl":"https://doi.org/10.1142/9789811200489_0001","url":null,"abstract":"","PeriodicalId":322478,"journal":{"name":"Algebraic and Geometric Combinatorics on Lattice Polytopes","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129698317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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