Is $C=\{x\in\mathbb{R}^n\; :\; \max \{x_1,x_2,x_3,\dots,x_n\}\leq 1\}$ convex or not?
As it has been mentioned in book of optimization that every max function on $\mathbb{R}^n$ is convex. So, I think for it may not effect whether it is equal to any positive or negative number. But I want to make sure, if my thinking is correct or not. Thank in advance.
Yes, it is convex.
Suppose $\max(x) \le 1$ and $\max(y) \le 1$. That is each component of $x$ and $y$ is at most $1$.
The $i$-th component is $\lambda x_i + (1-\lambda ) y_i \le \lambda + (1-\lambda)=1 $
Remark: In fact, it can be viewed as intersection of $n$ halfspaces.