Let $\Sigma$ be a fan in $N_{\mathbb{R}}$. A function $\Psi\colon|\Sigma|\to\mathbb{R}$ is called a virtual support function on $\Sigma$ if it is a conic $H$-lattice function (Definition 3.88). Alternatively, a virtual support function is a function $\Psi\colon|\Sigma|\to\mathbb{R}$ such that, for every cone $\sigma\in\Sigma$, there exists $m_{\sigma}\in M$ with $\Psi(u)=\langle m_{\sigma},u\rangle$ for all $u\in\sigma$. A set of functionals $\{m_{\sigma}\}_{\sigma\in\Sigma}$ as above is called a set of defining vectors of $\Psi$. A concave virtual support function on a complete fan will be called a support function.