A Conditional Upper Bound for the Moving Sofa Problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917853263822848 |
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| author | Baek, Jineon |
| author_facet | Baek, Jineon |
| contents | The moving sofa problem asks for the connected shape with the largest area $μ_{\text{max}}$ that can move around the right-angled corner of a hallway $L$ with unit width. The best bounds currently known on $μ_{\max}$ are summarized as $2.2195\ldots \leq μ_{\max} \leq 2.37$. The lower bound $2.2195\ldots \leq μ_{\max}$ comes from Gerver's sofa $S_G$ of area $μ_G := 2.2195\ldots$. The upper bound $μ_{\max} \leq 2.37$ was proved by Kallus and Romik using extensive computer assistance. It is conjectured that the equality $μ_{\max} = μ_G$ holds at the lower bound.
We develop a new approach to the moving sofa problem by approximating it as an infinite-dimensional convex quadratic optimization problem. The problem is then explicitly solved using a calculus of variation based on the Brunn-Minkowski theory. Consequently, we prove that any moving sofa satisfying a property named the injectivity condition has an area of at most $1 + π^2/8 = 2.2337\dots$. The new conditional bound does not rely on any computer assistance, yet it is much closer to the lower bound $2.2195\ldots$ of Gerver than the computer-assisted upper bound $2.37$ of Kallus and Romik. Gerver's sofa $S_G$, the conjectured optimum, satisfies the injectivity condition in particular. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_10725 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Conditional Upper Bound for the Moving Sofa Problem Baek, Jineon Metric Geometry Optimization and Control 49Q10, 52A10, 52A41 The moving sofa problem asks for the connected shape with the largest area $μ_{\text{max}}$ that can move around the right-angled corner of a hallway $L$ with unit width. The best bounds currently known on $μ_{\max}$ are summarized as $2.2195\ldots \leq μ_{\max} \leq 2.37$. The lower bound $2.2195\ldots \leq μ_{\max}$ comes from Gerver's sofa $S_G$ of area $μ_G := 2.2195\ldots$. The upper bound $μ_{\max} \leq 2.37$ was proved by Kallus and Romik using extensive computer assistance. It is conjectured that the equality $μ_{\max} = μ_G$ holds at the lower bound. We develop a new approach to the moving sofa problem by approximating it as an infinite-dimensional convex quadratic optimization problem. The problem is then explicitly solved using a calculus of variation based on the Brunn-Minkowski theory. Consequently, we prove that any moving sofa satisfying a property named the injectivity condition has an area of at most $1 + π^2/8 = 2.2337\dots$. The new conditional bound does not rely on any computer assistance, yet it is much closer to the lower bound $2.2195\ldots$ of Gerver than the computer-assisted upper bound $2.37$ of Kallus and Romik. Gerver's sofa $S_G$, the conjectured optimum, satisfies the injectivity condition in particular. |
| title | A Conditional Upper Bound for the Moving Sofa Problem |
| topic | Metric Geometry Optimization and Control 49Q10, 52A10, 52A41 |
| url | https://arxiv.org/abs/2406.10725 |