The sharp doubling threshold for approximate convexity

Journal article

We show for A , B ⊂ R d $A,B\subset \mathbb {R}^d$ of equal volume and t ∈ ( 0 , 1 / 2 ] $t\in (0,1/2]$ that if | t A + ( 1 − t ) B | < ( 1 + t d ) | A | $|tA+(1-t)B|< (1+t^d)|A|$ , then (up to translation) | co ( A ∪ B ) | / | A | $|\operatorname{co}(A\cup B)|/|A|$ 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.

Authors: van Hintum, P; Keevash, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Bulletin of the London Mathematical Society
• Publication date: 2024-07-25
• Acceptance date: 2024-06-19
• DOI: 10.1112/blms.13129
• EISSN: 1469-2120
• ISSN: 0024-6093 and 1469-2120
• Language: English