Hausdorff dimension of unions of $k$-planes

Abstract: We prove a conjecture of H\'era on the dimension of unions of $k$-planes. Let $0<k \le d<n$ be integers, and $\beta\in[0,k+1)$. If $\mathcal{V}\subset A(k,n)$, with $\text{dim}(\mathcal{V})=(k+1)(d-k)+\beta$, then $\text{dim}(\bigcup_{V\in\mathcal{V}}V)\ge d+\min{1,\beta}$. The proof combines a recent idea of Zahl and the Brascamp-Lieb inequality.