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 > 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 \(t > 0\)
• \(f\qty(x) = - \log x\) is convex, so relative entropy \(g\qty(x,t) = t \log t - t \log x\) is convex on \(\mathbb{R}_{++}^{2}\)