The Legendre transform of ##f(x) = \exp(\lvert x\rvert )##

[Wuberdall]

Homework Statement

Let the single variable real function f:\mathbb{R}\rightarrow\mathbb{R} be given by f(x)=e^{|x|}.
Determine the Legendre transform of f.

Homework Equations

Let I\subseteq\mathbb{R}be an interval, and f:I\rightarrow\mathbb{R}a convex function. Then its Legendre transform is the function f^{\ast}:I^{\ast}\rightarrow\mathbb{R}defined by : f^{\ast}(p) = \sup\lbrace xp - f(x)\hspace{1mm}\vert\hspace{1mm}x\in\mathbb{R}\rbrace.

The Attempt at a Solution

The function f is clearly a convex function and the supremum can easily by evaluated by finding the global maximum for xp-f(x). This yields

f^{\ast}(p) = \left\lbrace<br /> \begin{aligned}<br /> &amp;p\big(\ln p - 1\big) \hspace{6pt},\hspace{12pt}p&gt;0 \\<br /> &amp;-1 \hspace{6pt},\hspace{46pt}p=0\\<br /> &amp;f(-p) \hspace{6pt},\hspace{36pt}p&lt;0<br /> \end{aligned}<br /> \right.

My problem is that this function isn't convex nor isn't continuous on \mathbb{R} as I would expect the Legendre transform to be... Have I misunderstood something or simply done the legendre transform wrong? If the latter what have I then done wrong?

Thanks in advance.

 

[Ray Vickson]

Homework Statement

Let the single variable real function f:\mathbb{R}\rightarrow\mathbb{R} be given by f(x)=e^{|x|}.
Determine the Legendre transform of f.

Homework Equations

Let I\subseteq\mathbb{R}be an interval, and f:I\rightarrow\mathbb{R}a convex function. Then its Legendre transform is the function f^{\ast}:I^{\ast}\rightarrow\mathbb{R}defined by : f^{\ast}(p) = \sup\lbrace xp - f(x)\hspace{1mm}\vert\hspace{1mm}x\in\mathbb{R}\rbrace.

The Attempt at a Solution

The function f is clearly a convex function and the supremum can easily by evaluated by finding the global maximum for xp-f(x). This yields

f^{\ast}(p) = \left\lbrace<br /> \begin{aligned}<br /> &amp;p\big(\ln p - 1\big) \hspace{6pt},\hspace{12pt}p&gt;0 \\<br /> &amp;-1 \hspace{6pt},\hspace{46pt}p=0\\<br /> &amp;f(-p) \hspace{6pt},\hspace{36pt}p&lt;0<br /> \end{aligned}<br /> \right.

My problem is that this function isn't convex nor isn't continuous on \mathbb{R} as I would expect the Legendre transform to be... Have I misunderstood something or simply done the legendre transform wrong? If the latter what have I then done wrong?

Thanks in advance.

I suggest you plot y = exp(|x|) over some x-interval (-a,a) to clarify your thoughts about the function.