We first prove an exact result for hypergraphs: given $r\ge 2$, let $p$ be the smallest prime factor of $r-1$. If $n> (p-1)r$ and $G$ is an $r$- uniform hypergraph on $[n]$ such that every $r+1$ vertices contain $0$ or $r$ edges, then $G$ is either empty or a star, $\{E\subset [n]: |E|=r, E\ni x\}$ for some $x\in [n]$. Then we use it to slightly improve best known bounds for hypergraph Tur\'an numbers. We show that $\pi(K^r_{r+1})\leq 1- \frac{1}{r} - \left(1- \frac{1}{r^{p-1}}\right)\frac{(r-1)^2}{2r^p({r+p\choose p-1}+{r+1\choose 2})}$ when $r\ge 4$ is even. This is joint work with Linyuan Lu.