Geodesic
An integral estimate for deviations of sets in sections
Izvestiya. Mathematics , Tome 13 (1979) no. 2, pp. 261-276.

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Under certain natural assumptions it is proved that for two compact sets close in the Hausdorff metric the average distance between plane sections of them is also small. Bibliography: 4 titles. 
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G. Yu. Zaitsev. An integral estimate for deviations of sets in sections. Izvestiya. Mathematics , Tome 13 (1979) no. 2, pp. 261-276. http://geodesic.mathdoc.fr/item/IM2_1979_13_2_a3/

[1] Vitushkin A. G., “Dokazatelstvo polunepreryvnosti sverkhu variatsii mnozhestv”, Dokl. AN SSSR, 166:5 (1966), 1022–1026

[2] Ivanov L. D., Variatsii mnozhestv i funktsii, Nauka, M., 1975 | MR

[3] Meilanov V. S., “Dva blizkikh mnozhestva s ogranichennymi variatsiyami”, Matem. zametki, 19:4 (1976), 653–656 | MR | Zbl

[4] Federer H., Geometric measure theory, Springer, Berlin, 1969 | MR