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

For n≥ 2, we study the optimal function φₙ in the Brunn-Minkowski inequality for $θ$-convolution bodies |K+_θL|1/n≥φₙ(θ)(|K|1/n+|L|1/n) for any convex bodies K,Lⁿ. We improve the known lower bound for every 0<θ<(3/4)ⁿ and determine φₙ 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.