Variation of a set

A number characterizing the $k$- dimensional content of a set in $n$- dimensional Euclidean space. The zero variation ${V _ {0} } ( E)$ of a closed bounded set $E$ is the number of components of this set.

In the simplest case of the plane, the linear variation of a set $E$( i.e. the first-order variation of $E$) is the integral

$$V _ {1} ( E) = c \int\limits _ { 0 } ^ { {2 } \pi } \Phi ( \alpha , E) d \alpha$$

of the function

$$\Phi ( \alpha , E) = \int\limits _ {\Pi _ \alpha } V _ {0} ( E \cap \Pi _ {\alpha ,z } ^ \perp ) dz ,$$

where the integration is performed over the straight line $\Pi _ \alpha$ passing through the coordinate origin, $\alpha$ is the angle formed by $\Pi _ \alpha$ with a given axis and $\Pi _ {\alpha , z } ^ \perp$ is the straight line normal to $\Pi _ \alpha$ which intersects it at the point $z$. The normalizing constant $c$ is so chosen that the variation ${V _ {1} } ( E)$ of an interval $E$ is equal to its length. For sufficiently simple sets, e.g. for rectifiable curves, the variation of the set is equal to the length of the curve. For a closed domain $E$ with a rectifiable boundary $\Gamma$ its linear variation ${V _ {1} } ( E)$ is equal to one-half the length of $\Gamma$. The second variation of $E$( i.e. the second-order variation of $E$) is the two-dimensional measure of $E$, and ${V _ {k} } ( E) = 0$ if $k > 2$.

In $n$- dimensional Euclidean space the variation ${V _ {i} } ( E)$ of order $0 \dots n$, of a bounded closed set $E$ is the integral

$$V _ {k} ( E) = \ \int\limits _ {\Omega _ {k} ^ {n} } V _ {0} ( E \cap \beta ) d \mu _ \beta$$

of the zero variation of the intersection of $E$ with an $( n - k)$- dimensional plane $\beta$ in the space $\Omega _ {k} ^ {n}$ of all $( n - k)$- dimensional planes of $\mathbf R ^ {n}$ with respect to the Haar measure $d {\mu _ \beta }$; normalized so that the $k$- dimensional unit cube $J _ {k}$ has variation ${V _ {k} } ( J _ {k} ) = 1$.

The variation ${V _ {n} } ( E)$ is identical with the $n$- dimensional Lebesgue measure of the set $E$. For convex bodies the (suitably normalized) set variations are identical with Minkowski's mixed volumes (cf. Mixed-volume theory) [4].

Properties of the variations of a set.

1) The variations ${V _ {k} } ( E)$ for $E \subset \mathbf R ^ {n} \subset \mathbf R ^ {n ^ \prime }$ calculated for $E \subset \mathbf R ^ {n}$ and for $E \subset \mathbf R ^ {n ^ \prime }$ have the same value.

2) The variations of a set can be inductively expressed by the formula

$$\int\limits _ {\Omega _ {k} ^ {n} } V _ {i} ( E \cap \beta ) d \mu _ \beta = \ c( n, k, i) V _ {k+} i ( E),\ \ k+ i \leq n ,$$

where $c ( n, k, i)$ is the normalization constant.

3) ${V _ {i} } ( E) = 0$ implies ${V _ {i+} 1 } ( E) = 0$.

4) In a certain sense, the variations of a set are not dependent, i.e. for any sequence of numbers $a _ {0} \dots a _ {n}$, where $a _ {0}$ is a positive integer, $0 < a _ {i} \leq \infty$( $i = 1 \dots n - 1$), $a _ {n} = 0$, it is possible to construct a set $E \subset \mathbf R ^ {n}$ for which ${V _ {i} } ( E) = a _ {i}$, $i = 0 \dots n$.

5) If $E _ {1}$ and $E _ {2}$ do not intersect, $V _ {i} ( E _ {1} \cup E _ {2} ) = {V _ {i} } ( E _ {1} ) + {V _ {i} } ( E _ {2} )$. In the general case,

$$V _ {i} ( E _ {1} \cup E _ {2} ) \leq \ V _ {i} ( E _ {1} ) + V _ {i} ( E _ {2} ).$$

For $i = 0 \dots n - 1$ the variations $V _ {i}$ are not monotone, i.e. it can happen for $E _ {1} \supset E _ {2}$ that ${V _ {i} } ( E _ {1} ) < {V _ {i} } ( E _ {2} )$.

6) The variations of a set are semi-continuous, i.e. if a sequence of closed bounded sets $E _ {k}$ converges (in the sense of deviation in metric) to a set $E$, then

$$V _ {0} ( E) \leq \ {\lim\limits \inf } _ {k \rightarrow \infty } V _ {0} ( E _ {n} ) ,$$

and if, in addition, the sums ${V _ {0} } ( E _ {k} ) + \dots + {V _ {i-} 1 } ( E _ {k} )$ are uniformly bounded, then

$$V _ {i} ( E) \leq \ {\lim\limits \inf } _ {k \rightarrow \infty } V _ {i} ( E _ {k} ) ,\ \ i = 1 \dots n .$$

7) The variation ${V _ {k} } ( E)$ becomes identical with the $k$- dimensional Hausdorff measure if ${V _ {k+} 1 } ( E) = 0$ and if

$$V _ {0} ( E) + \dots + V _ {k} ( E) < \infty .$$

These conditions are met, for example, by twice-differentiable manifolds.

The concept of the variation of a set arose in the context of solutions of the Cauchy–Riemann system, and its ultimate formulation is due to A.G. Vitushkin. The set variations proved to be a useful tool in solving certain problems in analysis, in particular that of superposition of functions of several variables [1], and also in approximation problems [2].

References

 [1] A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian) [2] A.G. Vitushkin, "Estimation of the complexity of the tabulation problem" , Moscow (1959) (In Russian) [3] A.G. Vitushkin, "Proof of the upper semicontinuity of a set variation" Soviet Math. Dokl. , 7 : 1 (1966) pp. 206–209 Dokl. Akad. Nauk SSSR , 166 : 5 (1966) pp. 1022–1025 [4] A.M. Leontovich, M.S. Mel'nikov, "On the boundedness of the variations of a manifold" Trans. Moscow Math Soc. , 14 (1965) pp. 333–368 Trudy Moskov. Mat. Obshch. , 14 (1965) pp. 306–337 [5] L.D. Ivanov, "Geometric properties of sets with finite variation" Math. USSR-Sb. , 1 : 2 (1967) pp. 405–427 Mat. Sb. , 72 (114) : 3 (1967) pp. 445–470 [6] L.D. Ivanov, "On the local structure of sets with finite variation" Math. USSR-Sb. , 7 : 1 (1969) pp. 79–93 Mat. Sb. , 78 (120) : 1 (1969) pp. 85–100

Cf. also Content and Variation of a function.

How to Cite This Entry:
Variation of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_a_set&oldid=49116
This article was adapted from an original article by A.G. VitushkinL.D. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article