Estimating the surface area of spheres in normed spaces

The surface area of a polyhedron in a normed space is defined as the sum of the areas of its faces, each divided by the area of the central section of the unit ball, parallel to the face. This functional naturally extends to convex bodies.
In the paper, it is proved, in particular, that the surface area of the unit sphere in any three-dimensional normed space does not exceed 8.