ZONOIDS, AND BANG-BANG

The theory and practice of extremal problems,
the choice of optimal control in a deterministic and stochastic
environment, many techniques of mathematical economics rest
on the fundamental ideas of functional analysis which are connected
with convexity and measure.
The Lyapunov Convexity Theorem, proven in 1940
(see [1]–[5]),
occupies a prominent place in modern mathematics, since it lies at the juncture
of the theory of convex sets and measure theory.
The Lyapunov Convexity Theorem became the starting point
of numerous studies in the framework of mathematical analysis as
well as in the realm of geometric research into the convex sets that
are ranges of nonatomic vector measures.
The unexpectedness of the discovery by Lyapunov
is due to the paradoxical and fragile balance
of interactions between various finite- and infinite-dimensional ideas.
The effects of the Lyapunov Convexity Theorem vanish or disintegrate
in we admit into consideration nondiffuse (countably-additive) measures, or finitely-additive
measures, or measures with values in infinite-dimensional spaces
(e.g., see the second Lyapunov’s paper [2] and
[20]).
At the same time we should emphasize that, geometrically speaking,
the Lyapunov Convexity Theorem addresses the image under some linear operator
of the extreme points of a particular infinite-dimensional compact convex set.
This very circumstance was vital in the exquisite proof which was found by
Lindenstrauss in 1966 and which made
the Lyapunov Convexity Theorem very popular
(see [18]).
It is worth noting that today
there are available many proofs of the Lyapunov Convexity Theorem
that are grounded on the basic facts of mathematical analysis
(e.g., see [10] and
[12]).
For instance, such is
a rather elegant proof by Ross which was found in 2005 and bases
only on the Intermediate Value Theorem
(see [24]).
From the scratch the Lyapunov Convexity Theorem had raised the
problem of describing the compact convex sets in finite-dimensional real
spaces which serve as the ranges of diffuse measures.
These compacta are known in the modern geometrical literature as
* zonoids*. Among zonoids we distinguish the Minkowski sums of
finitely many straight line segments. These sets, called *zonotopes*,
fill a convex cone in the space of compact convex sets, and the cone of zonotopes
is dense in the closed cone of all zonoids.
The first description of the ranges of diffuse vector measures in the
Lyapunov Convexity Theorem was firstly found by
Chuikina practically in the modern terms
(see [6] and [7]).
Soon after that her result was somewhat supplemented and simplified by
Glivenko in [8].
The zonotopes of the present epoch were called
*parallelohedra* those days.
The significant further progress in studying the ranges of diffuse
vector measures belong to Reshetnyak and Zalgaller
who described zonoids as the results of mixing the linear elements
of a rectifiable curve in a finite-dimensional space in 1954
(see [9]).
In this same paper they suggested a new
prove of the Lyapunov Convexity Theorem and demonstrated that
zonotopes are precisely those convex polyhedra whose two-dimensional faces
have centers of symmetry. Unfortunately, these results remained
practically unnoticed in the West. Analogous results were
obtained by Bolker only fifteen years later in 1969
(see [11]).
We must mention the exceptional role of the Lyapunov Convexity Theorem
in justification of the “bang-bang” principle in the theory of optimal control.
The principle asserts that the optimal controls are implemented by
the extreme point of the set of admissible controls.
The meaning of the bang-bang principle is as follows:
For optimal transition in minimal time from one state of a system to the other
in the conditions of limited resources we can use an extreme “bang-bang”
control. In other words, if the system under control has an optimal control then
it has an optimal “bang-bang” control
(see [15, p. 47]).
For extra information
see, for instance,
[14], [16],
[17],[19],
and [21].
In closing we mention that the history of the Lyapunov Convexity Theorem
within functional analysis is displayed in some detail
in [23].
About the place of the theorem and search into its generalizations within measure theory
see [22].
As regards zonoids, see, e.g., [13].
##
References

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Ляпунов А. А.,
О вполне аддитивных вектор-функциях. I//
Изв. АН СССР, Сер. матем.,
**4**, 465–478 (1940).

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Ляпунов А. А.,
О вполне аддитивных вектор-функциях. II// Изв. АН СССР, Сер. матем.,
1946,
**10**, 277–279 (1946).

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**16**, Физ.-мат. ф-т,**3**, 97–126 (1951).

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Чуйкина К. И.,
Об аддитивных вектор-функциях.// Докл. АН СССР,
**76**, 801–804 (1951).

- [8]
Гливенко Е. В.,
О множествах значений аддитивных вектор-функций//
Матем. сб.,
**34**(76), 407–416, (1954).

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Залгаллер В.А., Решетняк Ю.Г.,
О спрямляемых кривых, аддитивных вектор-функциях и смешении отрезков//
Вестн. ЛГУ,
**2**, 45–65 (1954).

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Artstein Z.
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Proc. Amer. Math. Soc.,
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Bolker E.,
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Elton J., Hill Th.
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Halkin H.,
“A generalization of LaSalle’s bang-bang principle,”
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and Optimization,
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Hermes H., LaSalle J. P.,
*Functional Analysis and Time Optimal Control*. Academic Press, New York–London, 1969.

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Levinson N.,
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Lindenstrauss J.,
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Neustadt L. W.,
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Nunke R. J., Savage L. J.,
“On the set of values of a nonatomic,
finitely additive, finite measure,” Proc. Amer. Math. Soc.,
**3**:2, 217–218 (1952).

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Olech C.
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**2**, 74–101 (1966).

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Pap E. (Ed.)
*Handbook of Measure Theory.*Vol. 1 and 2. North Holland, Amsterdam (2002).

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Pietsch A.,
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- [24]
Ross D.,
“An elementary proof of Lyapunov’s theorem,” Amer. Math. Monthly
**112**:7, 651–653 (2005).

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