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\left(x,t\right) = t f\left(\frac{x}{t}\right), \text{dom } g = \left\{\left(x,t\right) \mid x / t \in \text{dom } f, t > 0\right\} \end{equation}
g is convex if f is convex. f\left(x\right) = x^{T}x is convex, so g\left(x,t\right) = x^{T}x / t is convex for t > 0 f\left(x\right) = - \log x is convex, so relative entropy g\left(x,t\right) = t \log t - t \log x is convex on \mathbb{R}_{++}^{2}