The sharp doubling threshold for approximate convexity

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-07-25 DOI:10.1112/blms.13129

Peter van Hintum, Peter Keevash

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Abstract

We show for $A, B \subset R^{d}$ of equal volume and $t \in (0, 1 / 2]$ that if $| t A + (1 - t) B | < (1 + t^{d}) | A |$ , then (up to translation) $| co (A \cup B) | / | A |$ is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by Figalli, van Hintum, and Tiba, the proof of which uses our current result. We additionally establish a similar sharp threshold for iterated sumsets.

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近似凸性的急剧加倍阈值

我们证明，对于 A , B ⊂ R d $A,B\subset \mathbb {R}^d$ 体积相等且 t∈ ( 0 , 1 / 2 ]$，如果 | t A + ( 1 - t ) 在 (0,1/2]$ 中，则 | t A + ( 1 - t ) = $。 $t\in (0,1/2]$ that if | t A + ( 1 - t ) B | < ( 1 + t d ) | A | $|tA+(1-t)B|< (1+t^d)|A|$ ，则（直至平移） | co ( A ∪ B ) | / | A | $|\operatorname{co}(A\cup B)|/|A|$ 是有界的。这就确立了菲加里、范欣图姆和蒂巴最近建立的布鲁恩-明考斯基不等式定量稳定性的尖锐阈值，其证明使用了我们当前的结果。我们还为迭代和集建立了类似的尖锐临界值。

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