Convex hulls of curves in $n$-space
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914971224375296 |
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| author | Scheiderer, Claus |
| author_facet | Scheiderer, Claus |
| contents | Let $K\subseteq{\mathbb R}^n$ be a convex semialgebraic set. The semidefinite extension degree ${\mathrm{sxdeg}}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of which is described by a linear matrix inequality of size $\le d$. This invariant can be considered to be a measure for the intrinsic complexity of semidefinite optimization over the set $K$. For an arbitrary semialgebraic set $S\subseteq{\mathbb R}^n$ of dimension one, our main result states that the closed convex hull $K$ of $S$ satisfies ${\mathrm{sxdeg}}(K)\le1+\lfloor\frac n2\rfloor$. This bound is best possible in several ways. Before, the result was known for $n=2$, and also for general $n$ in the case where $S$ is a monomial curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02359 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convex hulls of curves in $n$-space Scheiderer, Claus Algebraic Geometry Optimization and Control Let $K\subseteq{\mathbb R}^n$ be a convex semialgebraic set. The semidefinite extension degree ${\mathrm{sxdeg}}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of which is described by a linear matrix inequality of size $\le d$. This invariant can be considered to be a measure for the intrinsic complexity of semidefinite optimization over the set $K$. For an arbitrary semialgebraic set $S\subseteq{\mathbb R}^n$ of dimension one, our main result states that the closed convex hull $K$ of $S$ satisfies ${\mathrm{sxdeg}}(K)\le1+\lfloor\frac n2\rfloor$. This bound is best possible in several ways. Before, the result was known for $n=2$, and also for general $n$ in the case where $S$ is a monomial curve. |
| title | Convex hulls of curves in $n$-space |
| topic | Algebraic Geometry Optimization and Control |
| url | https://arxiv.org/abs/2410.02359 |