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Prove $$C := \{ x \in \mathbb R^n \mid \|x-c\| \leq a^T x + b \}$$ is a convex set.

I was trying to use the definition of a convex set, choosing two vectors in the set and getting two inequalities. But when I was trying to combine them, it came out with a constant term that cannot be canceled. Any help will be appreciated.

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  • $\begingroup$ Is the function $x \mapsto \|x-c\| -a^T x -b$ a convex function? $\endgroup$ – copper.hat Dec 5 '17 at 2:52
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Let $\varphi(x)=\|x-c\|-a^{T}x-b$, then \begin{align*} \varphi(tx+(1-t)y)&=\|tx+(1-t)y-c\|-a^{T}(tx+(1-t)y)-b\\ &=\|t(x-c)+(1-t)(y-c)\|-a^{T}(tx+(1-t)y)-tb-(1-t)b\\ &\leq\|t(x-c)\|+\|(1-t)(y-c)\|+t(-a^{T}x-b)+(1-t)(-a^{T}y-b)\\ &=t(\|x-c\|-a^{T}x-b)+(1-t)(\|y-c\|-a^{T}y-b)\\ &=t\varphi(x)+(1-t)\varphi(y). \end{align*}

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