  Katona's intersection theorem states that every intersecting family $\mathcal
F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal
F\vert$, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the
shadow of $\mathcal F$. Frankl conjectured that for $n>2k$ and every
intersecting family $\mathcal F\subseteq [n]^{(k)}$, there is some $i\in[n]$
such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$,
where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of
$\mathcal F$ at $i$. Here, we prove this conjecture in a very strong form for
$n> \binom{k+1}{2}$. In particular, our result implies that for any $j\in[k]$,
there is a $j$-set $\{a_1,\dots,a_j\}\in[n]^{(j)}$ such that $\vert \partial
\mathcal F(a_1,\dots,a_j)\vert\geq \vert\mathcal F(a_1,\dots,a_j)\vert$. A
similar statement is also obtained for cross-intersecting families.
