Hausdorff dimension of unions of $k$-planes

TL;DR

This paper proves a conjecture regarding the Hausdorff dimension of unions of k-planes, establishing a lower bound based on the dimension of the family of planes and employing advanced inequalities.

Contribution

It confirms Héra's conjecture on the dimension of unions of k-planes using innovative methods combining Zahl's idea and the Brascamp-Lieb inequality.

Findings

Established a lower bound for the dimension of unions of k-planes.

Validated a conjecture connecting the dimension of plane families to their unions.

Applied advanced inequalities to geometric measure theory problems.

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.