Fenchel conjugate of a pparticular function

by Juan-Pablo Contreras   Last Updated January 16, 2018 18:20 PM

I want to compute $H^*(w)$ for $H(v)=\frac{1}{2}|v|^2+|v_1|$ where $v$ is vectorial in $\mathbb{R}^d$, $|\cdot|$ reresents the euclidean norm and $v_1$ is the first component of $v$.

I know $H^*(w)=\sup_{v\in \mathbb{R}^d} v\cdot w - H(v)$. When norms are involved I use to write $v=|v|\cos \theta$ and say that the expresion is maximize when $\theta=\theta_{vw}$. But I don't know how to deal with this case when a single component apears.

Thanks for any help!

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