Intersection theoretic inequalities via Lorentzian polynomials
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910455602085888 |
|---|---|
| author | Hu, Jiajun Xiao, Jian |
| author_facet | Hu, Jiajun Xiao, Jian |
| contents | We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to $m$-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for $m$-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by $m$-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Plünnecke-Ruzsa inequalities) from additive combinatorics in our contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04191 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Intersection theoretic inequalities via Lorentzian polynomials Hu, Jiajun Xiao, Jian Algebraic Geometry Combinatorics Metric Geometry We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with respect to $m$-positive classes and Schur classes. We also study its convexity variants -- the geometric inequalities for $m$-convex functions on the sphere and convex bodies. Along the exploration, we prove that any finite subset on the closure of the cone generated by $m$-positive classes can be endowed with a polymatroid structure by a canonical numerical-dimension type function, extending our previous result for nef classes; and we prove Alexandrov-Fenchel inequalities for valuations of Schur type. We also establish various analogs of sumset estimates (Plünnecke-Ruzsa inequalities) from additive combinatorics in our contexts. |
| title | Intersection theoretic inequalities via Lorentzian polynomials |
| topic | Algebraic Geometry Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2304.04191 |