Virtual polytopes

Cyclopermutohedron: geometry and topology

“…Virtual polytopes. [4] Virtual polytopes appeared in the literature as useful geometrization of Minkowski differences of convex polytopes; below we give just a brief sketch. A convex polytope is the convex hull of a finite, non-empty point set in the Euclidean space R n .…”

“…Virtual polytopes. [4] Virtual polytopes appeared in the literature as useful geometrization of Minkowski differences of convex polytopes; below we give just a brief sketch.…”

“…The cyclopermutohedron is a virtual polytope, that is, the Minkowski difference of two convex polytopes. A detailed discussion on virtual polytopes can be found in the survey [4]. One of the messages of the survey is that virtual polytopes inherit almost all the properties and structures of convex polytopes: the volume (together with its polynomiality property), normal fan, face poset, etc.…”

“…One of the messages of the survey is that virtual polytopes inherit almost all the properties and structures of convex polytopes: the volume (together with its polynomiality property), normal fan, face poset, etc. However, virtual polytopes do not inherit the convexity property and therefore may appear as counter-intuitive: (1) The volume of a virtual polytope, although well-defined, can be negative, see [2,4]. The volume also can turn to zero, even if the virtual polytope does not degenerate.…”

“…The cyclopermutohedron CP 4 (computed in [3]) can be represented by a closed polygon, whereas its area (that is, two-dimensional volume) equals the integral of the winding number against the Lebesgue measure (see [4]). In other words, in this case the volume equals "sum of areas of six small triangles minus the area of the hexagon" in Fig.…”