  Let $\mathcal{L}$ be a family of lines and let $\mathcal{P}$ be a family of
$k$-planes in $\mathbb{F}^n$ where $\mathbb{F}$ is a field. In our first result
we show that the number of joints formed by a $k$-plane in $\mathcal{P}$
together with $(n-k)$ lines in $\mathcal{L}$ is
$O_n(|\mathcal{L}||\mathcal{P}|^{1/(n-k)}$). This is the first sharp result for
joints involving higher-dimensional affine subspaces, and it holds in the
setting of arbitrary fields $\mathbb{F}$. In contrast, for our second result,
we work in the three-dimensional Euclidean space $\mathbb{R}^3$, and we
establish the Kakeya-type estimate \begin{equation*}\sum_{x \in J}
\left(\sum_{\ell \in \mathcal{L}} \chi_\ell(x)\right)^{3/2} \lesssim
|\mathcal{L}|^{3/2}\end{equation*} where $J$ is the set of joints formed by
$\mathcal{L}$; such an estimate fails in the setting of arbitrary fields. This
result strengthens the known estimates for joints, including those counting
multiplicities. Additionally, our techniques yield significant structural
information on quasi-extremisers for this inequality.
