---
title: "perspective"
source: https://www.jemoka.com/posts/kbhperspective/
---

The perspective of a function: \(f: \mathbb{R}^{n} \to \mathbb{R}\) is the function: \(g: \mathbb{R}^{n} \times \mathbb{R} \to \mathbb{R}\):
\begin{equation} g\qty(x,t) = t f\qty(\frac{x}{t}), \text{dom } g = \qty {\qty(x,t) \mid x / t \in \text{dom } f, t &gt; 0} \end{equation}
\(g\) is convex if \(f\) is convex.
\(f\qty(x) = x^{T}x\) is convex, so \(g\qty(x,t) = x^{T}x / t\) is convex for \(t &gt; 0\) \(f\qty(x) = - \log x\) is convex, so relative entropy \(g\qty(x,t) = t \log t - t \log x\) is convex on \(\mathbb{R}_{++}^{2}\)
