Table Of ContentA.D. ALEXANDROV
SELECTED WORKS
Parti
Classics of Soviet Mathematics
Volume 1
L.S. PONTRYAGIN: SELECTED WORKS
1 Selected Research Papers
2 Topological Groups
3 Algebraic and Differential Topology
4 The Mathematical Theory of Optimal Processes
Edited by R. V. Gamkrelidze
Volume 2
N.N. BOGOLUBOV: SELECTED WORKS
Part I Dynamical Theory
Edited by N.N. Bogolubov Jr. and A.M. Kurbatov
Part II Quantum and Classical Statistical Mechanics
Edited t ~ olubov Jr., A.M. Kurbatov and A.S. Shumovsky
Part III fechanics and Pure Mathematics
Edited by V.S. Vladimirov
Part IV Quantum Field Theory
Edited by N.N. Bogolubov Jr.
Volume 3
L.V. KANTOROVICH: SELECTED WORKS
Part I Descriptive Theory of Sets and Functions. Functional Analysis in
Semi-ordered Spaces
Edited by S.S. Kutateladze
Part II Applied Functional Analysis. Approximation Methods and Computers
Edited by S.S. Kutateladze and J. V. Romanovsky
Volume 4
A.D. ALEXANDROV: SELECTED WORKS
Part I Selected Scientific Papers
Edited by Yu. G. Reshetnyak and S.S. Kutateladze
Volume 5
I.G. PETROWSKY: SELECTED WORKS
Part I Systems of Partial Differential Equations and Algebraic Geometry
Edited by O.A. Oleinik
Part II Differential Equations and Probability Theory
Edited by O.A. Oleinik
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A.D. ALEXANDROV
SELECTED WORKS
Part I
Selected Scientific Papers
Edited, by
Yu. G. Reshetnyak and S.S. Kutateladze
Novosibirsk University, Russia
Translated from the Russian by
P.S.V. Naidu
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CONTENTS
Editors’ Preface vii
Chapter I On infinitesimal bendings of nonregular surfaces 1
Chapter elementary proof of the Minkowski and some
other theorems on convex polyhedra 19
Chapter III To the theory of mixed volumes of convex bodies.
Part I: Extension of certain concepts of the theory
of convex bodies 31
Chapter IV To the theory of mixed volumes of convex bodies.
Part II: New inequalities for mixed volumes and
their applications 61
Chapter V To the theory of mixed volumes of convex bodies.
Part III: Extension of two Minkowski theorems
on convex polyhedra to all convex bodies 99
Chapter VI To the theory of mixed volumes of convex bodies.
Part IV: Mixed discriminants and mixed volumes 119
Chapter VII A general uniqueness theorem for closed surfaces 145
Chapter VIII Uniqueness theorems for closed surfaces 149
Chapter IX On the area function of a convex body 155
Chapter X Intrinsic geometry of an arbitrary convex surface 163
Chapter XI Existence of a convex polyhedron and a convex
surface with given metric 169
Chapter XII On tiling a space with polyhedra 175
v
vi CONTENTS
Chapter XIII On a generalization of Riemannian geometry 187
Chapter XIV The Dirichlet problem for the equation
det || zy|| = <pizt.......z„, Z, xt, x } . Part I 251
Chapter XV A general method for dominating solutions of
the Dirichlet problem 273
Chapter XVI On the principles of relativity theory 289
Index 319
EDITORS’ PREFACE
The present edition is the first part of the selected works of Alexandr Danilovich
Alexandrov, one of our most brilliant geometers. His scientific contributions ad
dress a wide range of problems of modem mathematics and its applications.
At the beeinninc of this century, geometry was in a position to deal with ob
jects “ii ’ However, approaches to them were not prompted by the
available methods of differential geometry or by the methods for studying the
solvability of the Cauchy problem and boundary-value problems for partial dif
ferential equations that were propounded in the 19th century. The efforts of such
outstanding mathematicians as H. Minkowski, D. Hilbert, H. Weyl, et al. led
only to isolated results. At the same time, their works contained statements of
many important unresolved problems that anticipated the further development of
geometry “in the large” in this century. Major discoveries in the study of these
problems were due to Alexandrov, who solved many of the discipline’s hard and
enduring problems.
The work of Alexandrov on the theory of generalized Riemannian spaces is
closely related to the area of mathematical analysis that studies the concept of
weak (generalized) solution to a differential equation (see, for instance, the fa
mous articles by S.L. Sobolev and L. Schwartz).
Alexandrov is also responsible for some important results on the existence of
weak solutions to some nonlinear partial differential equations (the equations of
Monge-Ampfcre type). In the case considered by Alexandrov, the concept of weak
solution is not reducible to that of the theory of partial differential equations.
Alexandrov was the first to apply many of the tools and methods of the theory
of real functions and functional analysis that are now current in geometry. One
of his first geometric papers using such pioneering methods is the article in
Chapter I, which concerns the infinitesimal bendings of a surface of revolution.
Here, the concept of infinitesimal bending is proposed and necessary and suffi
cient conditions are given for a vector field on a surface to be a bending field
on an arbitrary surface rectifiable in the Lebesgue sense. Then Alexandrov
proves a rigidity theorem for a surface of revolution, which asserts the absence
vii
viii EDITORS’ PREFACE
of infinitesimal bendings other than motions. The subtlety of this result is that, in
studying the equations of infinitesimal bendings with classical tools, there
appears the second derivative of the function describing a meridian of the sur
face under consideration. If this function is an arbitrary convex function then
methods requiring its second derivative cannot be applied directly.
Chapter II contains a proof of a uniqueness theorem for convex polyhedra in
three-dimensional space. The author derives from it a proof of the Minkowski
theorem claiming that two convex polyhedra coincide or are translates of one
another provided that their parallel faces have the same area. The proof of the
uniqueness theorem is of purely synthetic nature. It is based on the idea underly
ing the proof given by Cauchy of his remarkable theorem claiming that convex
polyhedra composed of congruent similarly placed faces are congruent.
Chapters III-VI present parts of a single treatise published in 1937 and 1938
in Matematicheskii Sbomik. Chapter IX also relates to this treatise. The ob
tained results are now considered fundamental in the theory of convex bodies
and the p ction offers an opportunity to become better acquainted
with the thcuiy uy consulting its definitive source. A short survey of the contents
of the treatise is given in Chapter III. The main topic of the treatise is the theory
of mixed volumes of systems of convex bodies. The concept of mixed volume
was suggested by H. Minkowski. In work by Brunn and Minkowski some in
equalities were obtained between various mixed volumes. The purpose of the
treatise by Alexandrov that is reprinted in Chapters III—VI is the establishment
of some new inequalities between mixed volumes, derivation of various geomet
ric corollaries of the inequalities, and proofs of numerous uniqueness theorems
in the theory of convex bodies and theorems on extremal properties of a ball and
bodies circumscribed around a ball. (Alexandrov’s inequalities were indepen
dently obtained by Fenchel and Jessen.) This treatise is characterized by the pro
lific use of tools from functional analysis and the theory of real functions. These
include the countably additive set functions on the unit sphere, area functions,
curvature functions and some other functions introduced by Alexandrov. Of the
series of results in these chapters, we should also mention the inequalities be
tween mixed discriminants of positive-definite quadratic forms, which are estab
lished in Chapter VI and are analogous to those for mixed volumes. These re
sults belong principally to algebra. Some of their geometric applications are
given in Chapter VI. This chapter also contains the proofs of Alexandrov’s main
inequalities between mixed volumes, which are obtained by means of functional
analysis (the Hilbert method, as the author calls it).
In Chapters VII and VIII a general uniqueness theorem is proved for regular
closed convex surfaces in three-dimensional space. In Chapter VII the validity
of the theorem is established under certain hypotheses, which are modified in
Chapter VIII. The uniqueness theorem of Chapter II is an analog of the general
theorem applied to polyhedra.
EDITORS’ PREFACE ix
Alexandrov indicated a new direction of research in geometry, namely the
theory of nonregular Riemannian spaces. Central to Riemannian geometry is the
theory of curvature of a space. Alexandrov constructed a theory of nonregular Ri
emannian spaces satisfying a specific generalized boundedness condition on cur
vature. His articles devoted to the topic further develop the geometric concept of
space along the lines of the tradition stemming from Lobachevskii, Riemann and
E. Cartan. Thereby mathematics was enriched with new ideas. Chapters X, XI and
XII contain the articles by Alexandrov that reflect this side of his activities.
Chapters X and XI deal with the intrinsic geometry of arbitrary convex sur
faces in the space R3. A surface is considered as a metric space in which the
distance between points is defined as the greatest lower bound of the lengths of
curves on the surface joining the points. The challenge here is to find conditions
for an arbitrary metric space to be isometric with a convex surface. Alexandrov
introduces the concept of a metric of positive curvature. The conditions sought
are as follows: the metric space under consideration is to be a two-dimensional
manifol' ic of positive curvature. In particular, it is proved that a two-
dimens:________jld with metric of positive curvature and homeomorphic to
the sphere is isometric to a closed convex surface. This theorem provides a new
solution to the familiar Weyl problem, but is now stated in an essentially more
general form than that considered by Weyl himself. The solution given by
Alexandrov is purely geometric in nature. The crux of the argument is the exist
ence theorem for a closed convex polyhedron with a given development.
Chapter XII relates to Alexandrov’s research into mathematical problems of
crystallography. Here he considers the problem of tiling a space with polyhedra
of which only a finite number are geometrically distinct.
Chapter XIII is devoted to the study of metric spaces of curvature not greater
than K, with K a constant. A particular instance of such a space is a Riemannian
space whose sectional curvature at every point is at most K. The definition is
given of a space with curvature not greater than K and various properties of such
spaces are established.
A wide range of articles by Alexandrov are devoted to the theory of partial
differential equations. The starting point for this research is within geometry.
The present volume contains only two of Alexandrov’s papers reflecting this
side of his activities. In Chapter XIV an existence theorem is proven for a solu
tion of the Dirichlet problem for an equation of the form det || z.. || = <p (z,,
zn, z, x,,.... xn), where z is an unknown function, z. are its first derivatives and z
are second derivatives. The concept of solution is treated here in a generalized
sense, and is determined by geometric means. The proof of the main result is
based on the methods developed in the theory of convex bodies. Chapter XV
outlines a method for estimating solutions to second order partial differential
equations that relies upon geometric considerations connected with the convex
envelope of a solution.
