Editing Hilbert's fourth problem (section)

==Hilbert's fourth problem and convex bodies==
Hilbert's fourth problem is also closely related to the properties of [[Convex body|convex bodies]]. A convex polyhedron is called a '''zonotope''' if it is the [[Minkowski addition|Minkowski sum]] of segments. A convex body which is a limit of zonotopes in the Blaschke – Hausdorff metric is called a [[zonoid]]. For zonoids, the [[support function]] is represented by
{{NumBlk||<math display="block">h(x) = \int_{S^{n-1}} \left|\left\langle x, u\right\rangle\right| \partial \sigma (u), </math>|{{EquationRef|1}}}}
where <math>\sigma (u)</math> is an even positive [[Borel measure]] on a sphere <math>S^{n-1}</math>.

The Minkowski space is  generated by the Blaschke–Busemann construction if and only if the support function of the indicatrix has the form of (1), where <math>\sigma (u)</math> is even and not necessarily of positive Borel measure.<ref>{{cite journal
| last1=Alexander | first1=Ralph
| title=Zonoid theory and Hilbert fourth problem
| journal=[[Geometriae Dedicata]]
| volume=28
| issue=2
| date=1988
| pages=199–211
| doi=10.1007/BF00147451 | s2cid=119391326
| doi-access=}}</ref> The bodies bounded by such hypersurfaces are called '''generalized zonoids'''.

The octahedron <math>|x_1| + |x_2| + |x_3| \leq 1</math> in the Euclidean space <math>E^3</math> is not a generalized zonoid. From the above statement it follows that the flat metric of Minkowski space with the norm <math>\|x\| = \max\{|x_1|, |x_2|, |x_3|\}</math> is not generated by the Blaschke–Busemann construction.