A remark on discrete Brunn-Minkowski type inequalities via transportation of measure (Q6594747)

The classical Brunn-Minkowski inequality \N\[\N\mathrm{vol}(A+B)^{\frac{1}{n}}\geq \mathrm{vol}(A)^{\frac{1}{n}}+\mathrm{vol}(B)^{\frac{1}{n}} \N\]\Nsatisfied for non-empty Borel measurable sets \(A, B\subset{\mathbb{R}}^n\) is equivalent to \N\[\N\mathrm{vol}(\lambda A+(1-\lambda)B)\geq \mathrm{vol}(A)^{\lambda}\mathrm{vol}(B)^{1-\lambda}, \ \forall\lambda\in[0,1]. \N\]\NA functional form of Brunn-Minkowski inequality is known as the Prékopa-Leindler inequality \N\[\N\int_{{\mathbb{R}}^n}h(x)dx\geq\bigg(\int_{{\mathbb{R}}^n}f(x)dx\bigg)^{\lambda}\bigg(\int_{{\mathbb{R}}^n}g(x)dx\bigg)^{1-\lambda},\ \ \forall\lambda\in[0,1], \N\]\Nassuming that the functions \(f, g, h\) satisfy \N\[\Nh(\lambda x+(1-\lambda)y)\geq f(x)^\lambda g(y)^{1-\lambda}, \ \ \forall x,y\in{\mathbb{R}}^n, \lambda\in[0,1]. \N\]\NThis paper studies the variant discrete Brunn-Minkowski inequality in functional form. It is known [\textit{D. Halikias} et al., Ann. Fac. Sci. Toulouse, Math. (6) 30, No. 2, 267--279 (2021; Zbl 1477.52014)] that every translation equivariant operation \(T: {\mathbb{Z}}^n\times{\mathbb{Z}}^n\to{\mathbb{Z}}^n\) which is monotone in the sense of Knothe admits a (discrete) Brunn-Minkowski inequality. \N\NIn this paper, the author shows that for \(\alpha, \beta,\gamma,\delta>0\) satisfying \(\max\{\alpha,\beta\}\leq\min\{\gamma,\delta\}\), and translation equivariant operation \(T: {\mathbb{Z}}^n\times{\mathbb{Z}}^n\to{\mathbb{Z}}^n\) which is monotone in the sense of Knothe, there holds a discrete Brunn-Minkowski type inequality \N\[\N\bigg(\sum_{x\in{\mathbb{Z}}^n}f(x)\bigg)^{\alpha}\bigg(\sum_{x\in{\mathbb{Z}}^n}g(x)\bigg)^{\beta}\leq\bigg(\sum_{x\in{\mathbb{Z}}^n}h(x)\bigg)^{\gamma}\bigg(\sum_{x\in{\mathbb{Z}}^n}k(x)\bigg)^{\delta} \N\]\Nfor four functions \(f,g,h,k: {\mathbb{Z}}^n\to[0,\infty)\) satisfying \N\[\Nf^\alpha(x)g^\beta(y)\leq h^\gamma(T(x,y))k^\delta(x+y-T(x,y)), \ \ \forall x,y\in{\mathbb{Z}}^n.\N\]