Improved Brunn--Minkowski inequality for $θ$-convolution bodies

For $n\geq 2$, we study the optimal function $φ_n$ in the Brunn-Minkowski inequality for $θ$-convolution bodies $|K+_θL|^{1/n}\geqφ_n(θ)(|K|^{1/n}+|L|^{1/n})$ for any convex bodies $K,L\subset\mathbb{R}^n$. We improve the known lower bound for every $0<θ<(3/4)^n$ and determine $φ_n$ on an interval strictly containing $[(2/3)^n,1]$. For $θ<1$ in this interval, equality holds if and only if $K$ and $-L$ are translates of the same $n$-dimensional simplex. The proof reduces the geometric problem to a sharp one-dimensional estimate for nonnegative concave functions under a weighted integral constraint.