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I want to prove that a ball for infinity norm is convex:

$$ B_\infty=\{x\in\mathbb R^n : \|x\|_\infty\le1\} $$

I came up with this proof and appreciate it if someone can help to verify if this is correct: \begin{align} \|x\|_\infty&=\|(1-\lambda)x+\lambda y\|_\infty\\&=\max_j|(1-\lambda)x|+\max_j|\lambda y|\\&=(1-\lambda)\max_j|x|+\lambda\max_j|y|\\&=(1-\lambda)+\lambda=1 \end{align}

P.S. This is a homework. Thanks!

Royi
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dresden
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    The overall concept seems fine except the left-hand side of your equation. It should be replaced by the right-hand side of the first line. Also, you might want to put subscripts for components of $x$ and $y$, and replace many occurrences of $=$ with $\le$. – Tunococ Oct 27 '13 at 12:58
  • Note: A ball induced by any norm is convex. This might be a tad easier to prove as you don't need to handle the maximum operator. – Tomas Oct 27 '13 at 13:15
  • Thank you all for your contributions, Tunococ for the answer and Donkey_2009 for edition. I very much appreciate it. I would like to also make a correction, for the previous comment from @Tomas that a ball induced with any norm greater than 1 will be convex. This is because of the triangle inequalities will be reversed when p-norm is between 0 and 1. Please correct if I am wrong. – dresden Oct 27 '13 at 15:34
  • @dracc What Tomas said was actually correct. When $p < 1$, the function you get is not a norm because, by definition, a norm has to satisfy the triangle inequality. – Tunococ Oct 27 '13 at 21:55

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Suppose $x, y \in B_{\infty}$. Let $x_i$ and $y_i$ denote components of $x$ and $y$. Then for any $\lambda \in [0, 1]$, \begin{align} \|(1 - \lambda)x + \lambda y\|_\infty & = \max_i |(1 - \lambda)x_i + \lambda y_i| & & \text{by definition of $\|\cdot\|_\infty$}\\ & \le \max_i (1 - \lambda)|x_i| + \max_i \lambda|y_i| & & \text{by $|a + b| \le |a| + |b|$} \\ & = (1 - \lambda)\|x\|_\infty + \lambda\|y\|_\infty & & \text{by definition of $\|\cdot\|_\infty$}\\ & \le (1 - \lambda) + \lambda & & \text{because $x \in B_\infty$ and $y \in B_\infty$}\\ & \le 1 \\ \therefore (1 - \lambda)x + \lambda y & \in B_{\infty} \end{align} Therefore, $B_{\infty}$ is convex.

Tunococ
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