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

Comments

Cf. also Content and Variation of a function.