$h$ is a support function of unit sphere $S^n$ in $\mathbb{R}^{n+1}$. Assume $h_{\max}$ is attained at the north pole. By convexity we have $$h(x) \geq h_{\max}x_{n+1}\quad \text{for}\quad x_{n+1}>0.$$ I wonder how this inequality come out. Thanks in advance!
*Update
Let's think of the one dimension case. In this case, $h$ is a support function of unit circle $S^1$ in $\mathbb{R}^2$,and we denote as $u$. Therefore, the inequality become
$$u(r\cos\theta,r\sin\theta)\geq u_{\max}\cdot(\sin\theta)_+$$
in polar coordinates, and $r=1$ since $u$ is defined on $S^1$. As we mentional above, $u_{\max}$ is attained at the north pole. So according to the definition of the support function, we have
$$u_{\max}=u(e_2)=\sup_{\theta}\{(0,1)\cdot (\cos\theta,\sin\theta)\}=1$$
and
$$u=\sup_{\theta}\{(r\cos\theta,r\sin\theta)\cdot(\cos\theta,\sin\theta)\}=\sup_{\theta}\{r\cos^2\theta+r\sin^2\theta\}=r=1.$$
For $(\sin\theta)_+\in[0,1]$, we have $u\geq u_{\max}\cdot(\sin\theta)_+.$
*Here is my thought, if there is some thing wrong please point it out, I will be very appreciated.