Assume that $ f:\mathbb{R}^n\to[0,+\infty) $ is a convex function and $ f(0)=0 $. It is obvious that $ 0 $ is the minimal point of $ f $. I want to ask if there exists $ \delta>0 $ such thta $ f(x)\leq C|x| $ for some $ C>0 $ if $ |x|<\delta $.
It is easy to show that as $ n=1 $, wuch $ C $ and $ \delta $ exist. However for $ n\geq 2 $, things may be a little bit complicated. Can you give me some hints or references?