This question is motivated by an obvious formal analogy between two well-known inequalities:

Log-concavity and Brunn-Minkowski inequality

Let $\mu(dx) := m(x) dx$ be an absolutely continuous measure on $\mathbb{R}^d$, such that its density $m$ is log-concave, i.e. for all $x,y\in \mathbb{R}^d, \lambda \in [0,1]$ $$m(\lambda x + (1-\lambda) y) \ge (m(x))^\lambda (m(y))^{1-\lambda}$$ Then for all measurable $X,Y \subset \mathbb{R}^d, \lambda \in [0,1]$ $$\mu(\lambda X + (1-\lambda) Y) \ge (\mu(X))^\lambda (\mu(Y))^{1-\lambda}$$ where $X$ and $Y$ are measurable, and $\lambda X + (1-\lambda) Y$ is a properly defined measurable version of $\{\lambda x + (1 - \lambda) y \mid x \in X, y \in Y\}$.

Note that the pointwise inequality for densities may be thought of as the Brunn-Minkowski inequality for, say, small enough balls.

Log-supermodularity and Ahlswede-Daykin inequality

Let $L$ be a finite or countable distributive lattice, and let $\mu_1,\dots,\mu_4$ be measures on $L$, such that for all $x,y \in L$ $$ \mu_1\{x \wedge y\} \mu_2\{x \vee y\} \ge \mu_3\{x\} \mu_4\{y\}$$ Then for all $X, Y \subset L$ $$ \mu_1(X \wedge Y) \mu_2(X \vee Y) \ge \mu_3(X) \mu_4(Y) $$

So here is the question: what are the other examples of inequalities that share a similar structure? A more precise question may be: what are other classes of $n$-ary operations $f_1,\dots,f_k$ on some space $S$, such that for some fixed $\lambda_i, \kappa_j \ge 0$, $\sum_i \lambda_i = \sum_j \kappa_j$ an inequality $$(\mu(Y_1))^{\lambda_1} \dots (\mu(Y_k))^{\lambda_k} \ge (\mu(X_1))^{\kappa_1} \dots (\mu(X_n))^{\kappa_n},$$ $$Y_i := f_i(X_1,\dots,X_n), X_j \subset S$$ follows from an "infinitesimal version" of itself - i.e. for sets that are in an appropriate sense "small enough"?

P.S. As I just learned from Wikipedia, Ahlswede-Daykin inequality has a nice generalization to $2k$ measures, due independently to Aharoni & Keich and Rinott & Saks that fits into the same framework.