To the problem of approximation by polyhedrons
of a convex surface with prescribed integral curvature
O. I. Trubina (Russia)
Vladimir State University
noi@vpti.vladimir.ru
3
We discuss the construction in Euclidean space E of the closed convex surface
P with prescribed integral curvature (see [1]). The existence of such surface is
proved in [1], where P is obtained as a limit of a polyhedrons sequence.
3
x
be the unit sphere centered at the origin O ; denote by dx
E
: x
1
Let S
the standard measure on S . Suppose that for any Borel set H  S the value of
integral curvature transferred from surface P to S is equal to
g
H
xdx
,
(1)
H
S, g0. Moreover, we assume that

C
where g is the given function, g
 H  satisfies the conditions of Aleksandrov’s theorem (see [1]).
We construct the special sequence of polyhedrons. For this purpose the sphere
T
is subjected to a finite triangulation S 
, such that all T are equal spherical

equilateral triangles. For rays l issuing from the point O to the centers of
triangles T and for numbers   T  there exists a unique, up to a homothety,
convex polyhedron with vertices on the rays l and with corresponding curvatures
 at these vertices ([1]). Further, drawing the midlines in triangle T (for each
 ), we obtain the subsequent triangulation finer than the previous one. As above
we construct a new class of homothetic closed convex polyhedrons. And so on.
By Pn  denote the class of homothetic polyhedrons obtained at the nth step of this
process.
3
Let  . , .  be the Euclidean distance in E .
3
Theorem. Suppose P E be the closed convex surface, which is starlike
relative to the origin O , and its integral curvature is defined according to (1); then




there exists a sequence P1 , P2 , … Pn , … of polyhedrons, Pn Pn , such that
for an arbitrary point M  P and for its corresponding point M n , where

n is large enough.
M
,M
M
OM
P
n n holds, if

n
n , the inequality
2
Note, that the constant C is uniquely determined by the function g , by the
diameter of P and by the diameter d of the initial triangulation, that
g
;  is an absolute constant (148).

C
,diam
P
,d
is C
The proof is based on methods and results developed in theory of manifolds
C
with bounded curvature (MBC), in the sense of Alexandrov. In particular, we use
a representation of the metric on MBC by Chebyshev line element ([2]). We use
also the estimate of the surface deformation generated by a variation of its metric
([3]).
The result can be applied to some elliptic Monge-Amper equations on a
sphere ([4], [5]) to estimate the convergence rate of approximate solutions.
References
[1] Alexandrov A. D. Existence and uniqueness of a convex surface with a
given integral curvature, Dokl. Akad. Nauk USSR, 35 (1942), 143 – 147.
[2] Bakelman I.Ya. Chebyshev nets in manifolds of bounded curvature, Tr.
Math. Inst. Steklova, 76 (1965), 124 – 129.
English transl.: Proc. Steklov Inst. Math., v. 76, 154-160(1967), Zbl104,168.
[3] Volkov Yu.A.
Ukr. Geom. Sbornic, 5 – 6 (1968), 44 – 69.
[4] Oliker V.I. Hypersurfaces in Rn+1 with prescribed Gaussian curvature and
related equations Monge-Ampere type, Comm. Partial Diff. Equations, 9 (8)
(1984), 807 – 838.
[5] Trubina O.I. Existence and uniqueness theorems for some class of MongeAmpere equations, Matem. Zametki, 34 (1), (1983), 123 – 130.