\tilde{\varphi}(x,y) = y\varphi(\frac{x}{y}).
Moreover, under these conditions, the epigraph of the perspective function
\left\{(t,x,y)~:~t\geq y\varphi(\frac{x}{y})\right\}
(or, to be precise, its appropriate closure) is a convex cone. Here are some familiar examples:
- \varphi(x)=x^2. Then \tilde{\varphi}(x,y)=\frac{x^2}{y}, familiar to some as quad-over-lin. The epigraph of \tilde{\varphi}, described by ty\geq x^2, is the Lorentz cone (rescaled rotated quadratic cone).
- \varphi(x)=x^p. Then \tilde{\varphi}(x,y)=\frac{x^p}{y^{p-1}} and the epigraph of \tilde{\varphi}, described equivalently by t^{1/p}y^{1-1/p}\geq |x| is the 3-dimensional power cone (with parameter p).
- \varphi(x)=\exp(x). Then the epigraph t\geq y\exp(x/y) is the exponential cone.
- \varphi(x)=x\log(x). Then the epigraph t\geq x\log(x/y) is the relative entropy cone.
We bring this up in connection with a series of blogposts by Dirk Lorenz here and here. For a monotone increasing, convex, nonnegative \varphi with \varphi(0)=0 he defines a norm on \mathbb{R}^n via
\|x\|_\varphi=\inf\left\{\lambda>0~:~\sum_i\varphi\left(\frac{|x_i|}{\lambda}\right)\leq 1\right\},
and we can ask when the norm bound t\geq \|x\|_\varphi is conic representable. The answer is: if the epigraph of the perspective function \tilde{\varphi} is representable, then so is the epigraph of \|\cdot\|_\varphi. The reason is that the inequality
\sum_i\varphi\left(\frac{|x_i|}{t}\right)\leq 1
is equivalent to
\begin{array}{ll} w_i\geq |x_i| & i=1,\ldots,n,\\ s_i\geq t\varphi\left(\frac{w_i}{t}\right) & i=1,\ldots,n,\\ t=\sum_i s_i. & \end{array}
That covers for example \varphi(x)=x^2, \varphi(x)=x^p (p>1), \varphi(x)=\exp(x)-1, \varphi(x)=x\log(1+x) and so on.
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| Figure: smallest (\exp(x)-1)-Luxemburg-norm disk containing a random set of points. |
