UDC 513.34

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR A. V. POGORELOV

A GENERAL EXISTENCE THEOREM FOR CLOSED CONVEX POLYHEDRA

In the present note a general existence theorem will be proved for a closed convex polyhedron with faces of prescribed directions and with prescribed values of a monotone function on the faces.

Let \(\alpha\) be an arbitrary plane and \(\omega_\alpha\) a function defined on closed convex polygons lying in planes parallel to the plane \(\alpha\), satisfying the following conditions:

1. The function \(\omega_\alpha\) is positive and continuous.

2. The function \(\omega_\alpha\) is invariant under parallel translations. This means that if the polygons \(P\) and \(Q\) coincide under a parallel translation, then the values of the function \(\omega_\alpha\) on these polygons are equal.

3. The function \(\omega_\alpha\) is strictly monotone in the sense that if the polygon \(Q\) is a part of the polygon \(P\), then \(\omega_\alpha(Q) < \omega_\alpha(P)\).

4. If the polygon \(P\) varies in such a way that its area \(s(P) \to \infty\), then \(\omega_\alpha(P) \to \infty\). If the polygon \(P\) degenerates (into a segment), then \(\omega_\alpha(P) \to 0\).

Theorem 1. Let \(\alpha_1, \alpha_2, \ldots, \alpha_n\) \((n \geq 3)\) be a system of planes not parallel to a single straight line; let \(\omega_1, \omega_2, \ldots, \omega_n\) be a system of functions defined on polygons parallel to the planes \(\alpha_1, \alpha_2, \ldots, \alpha_n\), respectively, satisfying conditions 1–4.

Then, whatever the positive numbers \(\varphi_1, \varphi_2, \ldots, \varphi_n\) may be, there exists a closed convex polyhedron with \(2n\) faces parallel to the planes \(\alpha_1, \alpha_2, \ldots, \alpha_n\), and with values of the functions \(\omega_k\) on these faces equal to \(\varphi_1, \varphi_2, \ldots, \varphi_n\).

This polyhedron has a center of symmetry and is determined uniquely up to a parallel translation.

We begin with the proof of the existence of the center of symmetry and the uniqueness of the polyhedron. Let \(P\) be a polyhedron whose existence is asserted by the theorem. Since it has \(2n\) faces, and a polyhedron can have no more than two faces of a prescribed direction, it follows that the polyhedron \(P\) has exactly two faces for each direction \(\alpha_k\). Construct the polyhedron \(P^*\), symmetric to the polyhedron \(P\) with respect to an arbitrary point \(O\). Put the faces of the polyhedra \(P\) and \(P^*\) into correspondence, considering as corresponding the parallel faces with equally directed outer normals. The values of the functions \(\omega\) on the corresponding faces of the polyhedra \(P\) and \(P^*\) are equal; consequently, these faces cannot be fitted one into another by a parallel translation. By a well-known theorem of A. D. Aleksandrov [((1), p. 265)], the polyhedra \(P\) and \(P^*\) are equal and parallelly situated, i.e., they coincide under a parallel translation. Hence we conclude that the polyhedron \(P\) has a center of symmetry. The uniqueness of the polyhedron \(P\) is proved analogously.

We now prove the existence of the polyhedron. We shall call an arbitrary system of \(n\) positive numbers \(\varphi_1, \varphi_2, \ldots, \varphi_n\) an abstract polyhedron. If there exists a closed convex polyhedron with \(2n\) faces parallel to the planes \(\alpha_1, \alpha_2, \ldots, \alpha_n\), and with values of the functions \(\omega_k\) on the faces parallel to the planes \(\alpha_k\) equal to \(\varphi_k\), then we shall call it a realization of the abstract polyhedron \((\varphi_1, \varphi_2, \ldots, \varphi_n)\).

Let us now define two manifolds. The elements of the first manifold, which we shall denote by \(M_1\), are closed convex symmetric polyhedra with \(2n\) faces parallel to the planes \(\alpha_1, \alpha_2, \ldots, \alpha_n\). Each such polyhedron is determined by \(n\) positive numbers—the values of the support function in the directions perpendicular to the planes \(\alpha_k\). It can be represented by a point of \(n\)-dimensional Euclidean space with coordinates equal to the indicated values of the support function. The second manifold, which we shall denote by \(M_2\), consists of abstract polyhedra, and it may be interpreted as the interior of the first coordinate angle of \(n\)-dimensional Euclidean space with Cartesian coordinates \(\varphi_1,\ldots,\varphi_n\).

To each polyhedron from \(M_1\) there naturally corresponds a certain abstract polyhedron from \(M_2\). We assert that this correspondence is a homeomorphism, and consequently every abstract polyhedron admits a geometric realization. The proof of this homeomorphism will be based on the “mapping lemma” of A. D. Aleksandrov ((\(^{1}\)), p. 127), as applied to the manifolds \(M_1\) and \(M_2\). The conditions of this lemma are satisfied. Indeed, the manifolds \(M_1\) and \(M_2\) have the same dimension \((n)\). The manifold \(M_2\) is connected, as a convex set. There exist polyhedra that are certainly realizable. The images of distinct points of \(M_1\) in \(M_2\) are distinct by virtue of the uniqueness proved above (polyhedra that can be made to coincide by a parallel translation are identified). It remains to prove that if some abstract polyhedron is the limit of realizable ones, then it is itself realizable.

Let \(P_1, P_2, \ldots\) be an infinite sequence of polyhedra from \(M_1\); let \(P'_1, P'_2, \ldots\) be the sequence of corresponding abstract polyhedra converging to the abstract polyhedron \(P\). We shall show that the polyhedron \(P\) is realizable. Denote by \(h_1^k, h_2^k, \ldots, h_n^k\) the support numbers of the polyhedron \(P_k\) in the directions perpendicular to the planes \(\alpha_1, \alpha_2, \ldots, \alpha_n\). Without loss of generality, we may assume that each sequence \(H_s(h_s^1, h_s^2, h_s^3, \ldots)\), \(s=1,\ldots,n\), converges to a finite or infinite limit.

We assert that each sequence \(H_s\) converges to a finite limit different from zero. Indeed, the surface areas of the polyhedra \(P_k\) are bounded in the aggregate. Therefore, if some sequence \(H_s\) converges to an infinite limit, then for sufficiently large \(k\) the polyhedron \(P_k\) is contained in a cylinder of arbitrarily small radius. In that case, among the faces of the polyhedron there will be one whose diameter is of the order of the diameter of the cylinder, and hence the function \(\omega\) for this face tends to zero as \(k \to \infty\), which is impossible.

If we now suppose that some sequence \(H_s\) converges to zero, then for sufficiently large \(k\) the polyhedron \(P_k\) lies between two arbitrarily close parallel planes. In that case there will be a face which, as \(k \to \infty\), degenerates into a segment; that is, again the function \(\omega\) tends to zero, which is impossible.

Thus all sequences \(H_s\) have positive limiting values, and hence it follows that the polyhedra \(P_k\) converge to a realization of the limiting polyhedron of the sequence \(P'_k\). Now, on the basis of the above-mentioned lemma of A. D. Aleksandrov, we conclude that the manifolds \(M_1\) and \(M_2\) are homeomorphic, and consequently that all abstract polyhedra are realizable. The theorem is proved.

As a consequence of Theorem 1 we obtain the following.

Theorem 2. Let \(\alpha_1, \alpha_2, \ldots, \alpha_n\) \((n \ge 3)\) be a system of planes no two of which are parallel to one line; let \(f(s,p,v_k)\) be a function of the positive variables \(s,p\) and of the unit vectors \(v_k\) perpendicular to the planes \(\alpha_k\), satisfying the conditions: 1) the function \(f\) is positive, continuous, strictly increasing in the variables \(s,p\), and even in \(v\) \((f(s,p,v)=f(s,p,-v))\); 2) as \(s \to \infty\), \(f(s,p,v)\to\infty\), and as \(s \to 0\), \(f(s,p,v)\to 0\).

Then, whatever the even positive function \(\varphi(\mathbf v_k)\) may be, there exists a closed convex polyhedron with \(2n\) faces, parallel to the planes \(a_k\), such that the areas \(s_k\), perimeters \(p_k\), and outer normals \(\mathbf v_k\) of its faces satisfy the conditions

\[ f(s_k, p_k, \mathbf v_k) = \varphi(\mathbf v_k). \]

This polyhedron has a center of symmetry and is determined uniquely up to a parallel translation.

Physical-Technical Institute of Low Temperatures
Academy of Sciences of the USSR

Received
16 I 1967

REFERENCES

1. A. D. Aleksandrov, Convex Polyhedra, Moscow–Leningrad, 1950.