Virtual polytopes have a long history, going back at least as far as Aleksandrov’s proof of the quadratic inequalities for mixed volumes, but were first formally named by A. V. Pukhlikov and A. G. Khovanskii [St. Petersbg. Math. J. 4, No. 2, 337–356 (1993); translation from Algebra Anal. 4, No. 2, 161–185 (1992; Zbl 0791.52010)]. The Minkowski sum \[ K \otimes L = \{x + y \mid x \in K,\;y \in L\} \] of \(K,L \in \mathcal{P}\), the family of convex polytopes in euclidean space \(\mathbb{R}^d\), turns \(\mathcal{P}\) into a semigroup with cancellation. (The multiplicative notation is used because, as the authors point out, Minkowski addition behaves in some contexts much more like multiplication.) With translates (usually) identified, this semigroup embeds in the corresponding Grothendieck group \(\mathcal{P}^*\), whose elements are the virtual polytopes. A general element of \(\mathcal{P}^*\) is then written \(K \otimes L^{\otimes-1}\), with \(L^{\otimes-1}\) the inverse of \(L\) in \(\mathcal{P}^*\) with respect to \(\otimes\). Since a polytope is determined by its support function (which is piecewise linear and convex), a virtual polytope can be identified with a difference of support functions, factored out by the globally linear functions.
In this survey, after an initial discussion of how concepts such as faces and normal fans can be extended to \(\mathcal{P}^*\), the authors then look in detail at the \(2\)-dimensional case. A virtual polygon can be thought of as a (closed) \(2\)-coloured circuit of edges, with the colours indicating whether they are counted positively or negatively. In a similar way, the normal fan gives rise to a coloured star.
A polytope \(K\) can be identified with its characteristic function \(I_K\), defined by \[ I_K(x) = \begin{cases} 1, &\text{if } x \in K, \\ 0, & \text{if } x \notin K. \end{cases} \] These characteristic functions generate a ring, with the usual addition – which corresponds to the valuation property – and multiplication defined by convolution giving the Minkowski sum; initially at least, translations are not factored out. (The basic ideas here go back to H. Groemer [Geom. Dedicata 6, 141–163 (1977; Zbl 0394.52003)]). An important feature is that \[ (I_K)^{\otimes-1} = \sum_F \, (-1)^{\dim K - \dim F}I_F, \] with the sum over all faces \(F\) of \(K \in \mathcal{P}\) including \(K\) itself (this expression is not given in the paper – apart from the sign \((-1)^{\dim K}\), the right side is the characteristic function of the relative interior of \(K\)).
The connexions with the reviewer’s polytope algebra \(\Pi\) [Adv. Math. 78, No. 1, 76–130 (1989; Zbl 0686.52005)] are considered next. Except in a minor respect, \(\Pi\) is a graded algebra over \(\mathbb{R}\), since once again translates are identified; now, the first weight space of \(\Pi\) is isomorphic to \(\mathcal{P}^*\). The relationships with fans are then discussed.
The \(3\)-dimensional case is also treated in depth. Some of the notions in this case go back to Maxwell, with stressed graphs on the sphere; we shall not go into the details.
Finally, the authors consider applications. These are to Aleksandrov’s problem on the uniqueness of convex surfaces, as well as topics already mentioned such as valuations including volume and counting integer points in lattice polytopes, mixed volumes, decomposition of polytopes under Minkowski addition, and finally a brief look at projective toric varieties and associated subjects.