The authors give characterizations of convex sets or domains in \(\overline{\mathbb{C}}\) in terms of their conformal/hyperbolic radius \(R\) and in terms of their mappings by the gradient function \(\nabla{R}\). They use the complex-valued mapping \(\nabla{R}(\cdot, \Omega)\,:\, \Omega \longrightarrow \overline{\mathbb C}\) to give a unified proof and some generalizations of a number of known results on convexity of \(\Omega\) or \({\mathbb C}\setminus{\Omega}\) when the Jacobian \(J(\cdot, \Omega)\) of the gradient is non-negative. For example, for simply connected domains \(\Omega\) with \(J(\cdot, \Omega)\geq 0\), the convex domain \(\Omega\) is neither a half-plane nor a strip nor an angular domain \(\Leftrightarrow\) the hyperbolic radius \(R(\cdot, \Omega)\) is a strictly concave function \(\Leftrightarrow\) the function \(\nabla{R}(\cdot, \Omega)\) is a diffeomorphism of \(\Omega\) onto a domain \(G\) contained in the disk \(D_2=\{\zeta\,:\,| \zeta| <2\}\). This theorem characterizes a result by Kim, Minda, and Wright in terms of \(\nabla{R}(\cdot, \Omega)\) and contains a classic result by Löwner. The authors give similar characterizations for other types of domains, including doubly connected domains.