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 s