Variation of a set

A number characterizing the -dimensional content of a set in -dimensional Euclidean space. The zero variation of a closed bounded set is the number of components of this set.

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

of the function

where the integration is performed over the straight line passing through the coordinate origin, is the angle formed by with a given axis and is the straight line normal to which intersects it at the point . The normalizing constant is so chosen that the variation of an interval 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 with a rectifiable boundary its linear variation is equal to one-half the length of . The second variation of (i.e. the second-order variation of ) is the two-dimensional measure of , and if .

In -dimensional Euclidean space the variation of order , of a bounded closed set is the integral

of the zero variation of the intersection of with an -dimensional plane in the space of all -dimensional planes of with respect to the Haar measure ; normalized so that the -dimensional unit cube has variation .

The variation is identical with the -dimensional Lebesgue measure of the set . 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 for calculated for and for have the same value.

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

where is the normalization constant.

3) implies .

4) In a certain sense, the variations of a set are not dependent, i.e. for any sequence of numbers , where is a positive integer, (), , it is possible to construct a set for which , .

5) If and do not intersect, . In the general case,

For the variations are not monotone, i.e. it can happen for that .

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

and if, in addition, the sums are uniformly bounded, then

7) The variation becomes identical with the -dimensional Hausdorff measure if and if

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

Comments

Cf. also Content and Variation of a function.