By Carl Graham
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Then the limit surface F of the sequence of polyhedra P is smooth. In fact, the surface F has a bounded specific curvature. According to a theorem of A. D. Aleksandrov , a disturbance of the smoothness of a surface with bounded specific curvature can occur only along a recti linear segment with endpoints on the boundary of the surface. But the surface F is infinite. Therefore, disturbance of the smoothness can occur only along an entire line. And a convex infinite surface, containing at least one fine, is a cylinder.
J J 8'dpdq > J J pdxdy, G* 3. 5' = lim S '(x 9y 9z 9p 9q ). (xfy)eG Z->-oo G J J 3'dpdq < J J pdxdu, G* 3 G Z^O O (*,ÿ)6C ? ). Then there exists a generalized solution o f the equation S'(rt—s2)= p in the region G which satisfies the boundary condition \J/(p9q)= 09 and has convexity in the direction z < 0 . Now let the function S' depend only on p and q. We shall show that in this case Theorem 3 is valid if conditions 2 and 3 are replaced by the condition J G* q)dpdq = J J p'(x, y )d x d y . G We displace the cone V through the segment N in the direction z > 0 and we denote it in this position by Vn .
Let us assume that this is false; then we can construct a sequence of polyhedra P satisfying the conditions S(Ffc)^3* ( J c = l m), with vertices arbitrarily close to the point O. Without loss of generality, we can assume that the vertices of the polyhedra P converge, and, moreover, that at least one of these vertices converges to the point O. All of them cannot converge to the point O, inasmuch as We denote by V the convex hull of those rays gk along which the vertices of the polyhedra P do not converge to O.
Chaînes de Markov by Carl Graham