A convex approximation conjecture for near-minimizers of the Brunn–Minkowski inequality

Let $\Delta>0$ be an absolute constant. Convex approximation conjecture. If $\delta<\Delta$ and $A\subset\mathbb{R}^k$ satisfies

$$\left|\frac{A+A}{2}\right|\leq(1+\delta)|A|,$$

then there is a convex set $K\subset\mathbb{R}^k$ such that

$$|K\triangle A|\leq O_{\delta}(1)|A|.$$

This conjecture proposes a dimension-independent convex-structure conclusion for sets whose midpoint sum has nearly minimal measure, inspired by the one-dimensional functional stability theorem discussed in the source; it remains open according to the supplied evidence.