Signed Minkowski decompositions of convex polygons into minimum simplices and factorization of max-plus functions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909660316958720 |
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| author | Kitagawa, Soujun |
| author_facet | Kitagawa, Soujun |
| contents | Signed Minkowski decomposition is an expression of a polytope as a Minkowski sum and difference of smaller polytopes. Signed Minkowski decompositions of a polytope can be interpreted as factorizations of a max-plus (tropical) function. We review two relations about Minkowski decompositions, and we prove that any 2-dimensional integral polytopes (polygons) have a signed Minkowski decomposition which consists of integral points, integral line segments of length 1, and integral triangles of area 1/2. From this result, we also obtain that any max-plus functions with two variables and integer coefficients can be expressed as a set of specified form of simpler max-plus functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Signed Minkowski decompositions of convex polygons into minimum simplices and factorization of max-plus functions Kitagawa, Soujun Combinatorics 52C05, 52B20, 15A80 Signed Minkowski decomposition is an expression of a polytope as a Minkowski sum and difference of smaller polytopes. Signed Minkowski decompositions of a polytope can be interpreted as factorizations of a max-plus (tropical) function. We review two relations about Minkowski decompositions, and we prove that any 2-dimensional integral polytopes (polygons) have a signed Minkowski decomposition which consists of integral points, integral line segments of length 1, and integral triangles of area 1/2. From this result, we also obtain that any max-plus functions with two variables and integer coefficients can be expressed as a set of specified form of simpler max-plus functions. |
| title | Signed Minkowski decompositions of convex polygons into minimum simplices and factorization of max-plus functions |
| topic | Combinatorics 52C05, 52B20, 15A80 |
| url | https://arxiv.org/abs/2506.20999 |