Motivated by optimization considerations and the (matrix theory) inequalities of Ky Fan and von Neumann, we introduce a Fan-Theobald-von Neumann system as a triple $(V,W,\lambda)$, where $V$ and $W$ are real inner product spaces and $\lambda:V\rightarrow W$ is a (nonlinear) map satisfying the following condition: For all $c,u\in V$,
$$\max \{\langle c,x\rangle: x\in [u] \}=\langle \lambda(c),\lambda(u)\rangle,$$ where $[u]:= \{x:\lambda(x)=\lambda(u)\}$.

This simple formulation happens to be equivalent to the Fenchel conjugate formula of the form $(\phi\circ \lambda)^*=\phi^*\circ \lambda$ and a subdifferential formula in some settings and becomes useful in addressing linear/distance optimization problems over “spectral sets” which are of the form $\lambda^{-1}(Q)$, where $Q$ is a subset of $W$. Three standard examples of FTvN systems are: $(\mathbb{R}^n,\mathbb{R}^n,\lambda)$ with $\lambda(x):=x^\downarrow$ (the decreasing rearrangement of the vector $x\in \mathbb{R}^n$); $({\cal H}^n,\mathbb{R}^n,\lambda)$, where ${\cal H}^n$ is the space of $n$ by $n$ complex Hermitian matrices with $\lambda$ denoting the eigenvalue map; and $(M_n,\mathbb{R}^n, \lambda)$, where $M_n$ is the space of $n$ by $n$ complex matrices with $\lambda$ denoting the singular value map. Other examples come from Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). In the general framework of Fan-Theobald-von Neumann systems, we introduce and elaborate on the concepts of commutativity, automorphisms, majorization, etc. We will also talk about “transfer principles” where properties (such as convexity) of $Q$ are carried over to $\lambda^{-1}(Q)$, leading to a generalization of a celebrated convexity theorem of Chandler Davis.