The sharp doubling threshold for approximate convexity (Q6634560)

The Brunn-Minkowski inequality states that for sets \(A, B \subset \mathbb R^d\) with equal volume and \(t \in (0,1/2]\), we have \(|tA + (1-t)B| \ge |A|\), with equality if \(A=B\) is a convex set. These results provide various control (up to translation) \(|A \triangle B|\), \(|co(A) \setminus A|\), and \(|co(A \cup B) \setminus A|\) in terms of the parameter \(\delta_t(A,B) := \frac{|tA+(1-t)B|}{|A|} - 1 \ge 0\), where \(co(A)\) denotes the convex hull of \(A\), the smallest convex set containing \(A\).\N\NIt has been shown [\textit{A. Figalli} and \textit{D. Jerison}, Adv. Math. 314, 1--47 (2017; Zbl 1380.52010)] that there exist constants \(a_{d,t}, c_{d,t}, \Delta_{d,t} > 0\) such that if \(\delta := \delta_t(A,B) \le \Delta_{d,t}\), then (up to translation) \(|co(A \cup B) \setminus A| \le c_{d,t} \delta^{a_{d,t}}|A|\). For general \(A,B \subset \mathbb R^d\), the optimal values were determined to be \(a_{d,t} = 1/2\) and \(c_{d,t} = O_d(t^{-1/2})\) [\textit{A. Figalli} et al., ``Sharp quantitative stability of the Brunn-Minkowski inequality'', Preprint, \url{arXiv:2310.20643}]. For the special case where \(A=B\), a stronger result was established with \(a_{d,t}=1\) and \(c_{d,t} = t^{-1}\exp(O(d \log d))\) [\textit{P. van Hintum} et al., J. Eur. Math. Soc. (JEMS) 24, No. 12, 4207--4223 (2022; Zbl 1501.52007); \textit{A. Figalli} and \textit{D. Jerison}, Ann. Sci. Éc. Norm. Supér. (4) 54, No. 1, 235--257 (2021; Zbl 1482.11139)]. In this paper, the authors determine the optimal value of \(\Delta_{d,t}\) for these results, proving the following.\N\NTheorem 1.1. For all \(d \in \mathbb N\), \(t \in (0,1/2]\), there are \(C_{d,t}>0\) so that if \(A,B \subset \mathbb R^d\) of the same volume have \(|tA+(1-t)B| < (1+t^d)|A|\), then (after possibly translating) \(|co(A \cup B)| \le C_{d,t}|A|\).\N\NIn fact, it is possible to choose \(C_{d,t} = t^{-O(d^2)}\). The second theorem establishes this threshold for iterated sumsets. For \(X \subset \mathbb R^d\) and \(k \in \mathbb N\), define \(k \cdot X = \underbrace{X+\dots+X}_{k \text{ terms}}\).\N\NTheorem 1.2. For all \(d, k \in \mathbb N\), there exists \(C_{d,k}>0\) such that if \(A \subset \mathbb R^d\) satisfies \(|k \cdot A| < (1^d+\dots+k^d)|A|\), then \(|co(A)| \le C_{d,k}|A|\).\N\NThe following result provides a quantitative stability version of the Brunn-Minkowski inequality.\N\NCorollary 1.4. For all \(d \in \mathbb N\), \(t \in (0,1)\), there exist \(a_{d,t}, C_{d,t} > 0\) such that if \(A, B \subset \mathbb R^d\) have equal volume and satisfy \(\delta := \delta_t(A,B) < t^d\), then (up to translation) \(|co(A \cup B) \setminus A| \le C_{d,t}\delta^{1/2}|A|\), and \(|co(A) \setminus A| + |co(B) \setminus B| \le C_{d,t}\delta|A|\).\N\NAll these results are sharp, as demonstrated by the example \(A = [0,1]^d\) and \(B = A \cup \{v\}\), where \(v \in \mathbb R^d\) is an arbitrarily large vector. For these sets, \(tA + (1-t)B = A \cup ([0,1]^d + (1-t)v)\), so \(\delta_t(A,B) = t^d\), while \(\frac{|co(B) \setminus B|}{|B|} \to \infty\) as \(\Vert v \vert_2 \to \infty\).\N\NThe proofs rely on identifying two points in \(A\) (or \(B\)) for each coordinate direction that are significantly far apart, which can be done by increasing \(|co(A)|\). Two cases are then considered: either \(A\) contains long fibres in all coordinate directions, or it does not. In the first case, the authors find a lower bound on the doubling using Plünnecke's inequality as the sum of those long fibres is large. In the second case, it is fixed a direction in which the fibres of \(A\) are short and it is shown that using an optimal transport map, they pair up the fibres from \(A\) and \(B\) whose (weighted) sum form a reference set of size \(|A|\). Finally, it is shown that summing fibres of \(B\) with the two far removed points from \(A\) gives a set disjoint from the reference set of the required size.