Subdividing a Convex Body by a System of Cones and Polytopes Inscribed in the Body
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SUBDIVIDING A CONVEX BODY BY A SYSTEM OF CONES AND POLYTOPES INSCRIBED IN THE BODY V. V. Makeev∗ and N. Yu. Netsvetaev∗
UDC 514.172
The literature contains quite a few theorems on subdividing the volume of a convex body by a system of cones and on the possibility to circumscribe the body about a polytope of one type or another. See R. N. Karasev, “Topological methods in combinatorial geometry,” Russian Math. Surveys, 63, No. 6, 1031–1078 (2008) for a survey of similar results. In the following, we also prove theorems of this kind. As a limit case, we obtain well-known theorems on inscribed polytopes. Bibliography: 4 titles.
1. Systems of cones Notation. Everywhere below, by a convex body K in Rn we mean a compact convex set with nonempty interior, and V (K) denotes the volume of K. In what follows, we use the wellknown Hausdorff metric on the set of convex compacta in Rn (see [2, p. 8]). Furthermore, F denotes a nonnegative functional that is continuous with respect to the Hausdorff metric. Moreover, we assume that F (K) = 0 only if dim(K) < n. In addition to the volume V , which was mentioned above, there exist many functionals F that are natural from the geometric viewpoint: the smallest width, the largest volume of a ball or an ellipsoid contained in a convex compact set, the difference between the surface area and twice the maximal area of the orthogonal projection to a hyperplane, etc. Theorem 1. Let K be a smooth convex body, let F be a functional of the type described above, and let C = {C1 , . . . , Cn+1 } be a system of n + 1 nonoverlapping convex cones with nonempty interiors and common vertex A. We assume that each closed half-space bordered by a hyperplane passing through A entirely contains one of the cones Ci . If x ∈ K, we can translate the system C in parallel by shifting the common vertex A to the point x. Then there exists an x such that the values of F on the intersections of K with the shifted cones form the preassigned proportion. Proof. We define Δ = {(x1 , . . . , xn+1 ) ∈ Rn+1 | x1 + · · · + xn+1 = 1 and xi ≥ 0 for 1 ≤ i ≤ n + 1} and consider the continuous mapping f : K → Δ defined as follows. If x ∈ K, then, translating the cones C1 , . . . , Cn+1 in parallel, we can shift their common }. We define vi = F (Ci ∩ K) vertex A to x, thus obtaining a new system Cx = {C1 , . . . , Cn+1 and set v vn+1 1 ,..., . v = v1 + · · · + vn+1 and f (x) = v v Let us show that f is surjective. We observe that f (∂K) ⊂ ∂Δ. Indeed, our assumption about the cones C1 , . . . , Cn+1 implies that for each point x ∈ ∂K one of the coordinates of f (x) vanishes, whereas another one is necessarily positive. Consequently, there arises a mapping φ : (K, ∂K) → (Δ, ∂Δ) of pairs. Let us show that the degree of φ is equal to 1 modulo 2. ∗ St.Petersburg State University, [email protected].
St.Petersburg,
Russia,
e-mail:
[email protected],
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 476, 2018, pp. 125–130. Original article submitted June 18, 2018. 512 1072-3374/20/2514-0512 ©2020 Springer Science+Bus
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