Is $C=\{x\in\mathbb{R}^n\; :\; \max \{x_1,x_2,x_3,\dots,x_n\}\leq 1\}$ convex or not?

Question

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.

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Siong Thye Goh · Answer 1 · 2019-10-16T07:14:42.710

1

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.

edited Oct 16 '19 at 07:14

answered Oct 16 '19 at 07:08

Siong Thye Goh

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Is $C=\{x\in\mathbb{R}^n\; :\; \max \{x_1,x_2,x_3,\dots,x_n\}\leq 1\}$ convex or not?

1 Answers1