The fractional Brunn–Minkowski inequality for hypergraph fractional partitions

Let $M\geq2$, let $K_1,\ldots,K_M$ be nonempty Borel subsets of $\mathbb{R}^n$, and let $\mathcal{G}$ be a collection of subsets of $[M]$. A fractional partition using $\mathcal{G}$ is a collection of nonnegative weights $\beta_{\mathbf{s}}$, $\mathbf{s}\in\mathcal{G}$, such that

$$\sum_{\mathbf{s}\in\mathcal{G}:j\in\mathbf{s}}\beta_{\mathbf{s}}=1\quad\text{for every }j\in[M].$$

The fractional Brunn–Minkowski conjecture. For every such fractional partition,

$$|K_1+\cdots+K_M|^{1/n}\geq\sum_{\mathbf{s}\in\mathcal{G}}\beta_{\mathbf{s}}\left|\sum_{j\in\mathbf{s}}K_j\right|^{1/n}.$$

This extends the classical Brunn–Minkowski inequality by replacing the individual summands with sums indexed by a fractional partition. Its validity is proposed as a generalization of the convolution inequality, while the source does not establish the full assertion.