On area of 2-spheres in
Euclidean space

I.Ya. Oleksiv

It is proved that for a 2-sphere embedded in Euclidean 3-space the infimum of the area limits of approximating PL 2-spheres is equal to the Lebesgue area of the 2-sphere.

1. L. Cesari, Surface Area, vol. 35, (Annals of Mathematics Study), Princeton University Press, 1956, 596 pp., pp. 596.

2. R.H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. 65 (1957), 456–483.

3. R.P. Goblirsch, An area for simple surfaces, Ann. of Math. 68 (1958), 231–246.

4. И.Я. Олексив, О площади простой поверхности, ДАН СССР 221 (1975), no. 1, 30–31.

5. А.С. Кронрод, О функциях двух переменных, Успехи матем. наук 5 (1950), no. 1(35), 24–134.

6. K. Kuratowski, Sur les coupures irreductible du plan, Fund. Math. 6 (1924), 130–145.

7. W.H. Meeks, S.-T. Yau, The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglass-Morrey and the analytic proof of Dehn’s lemma, Topology 21 (1982), 409–442.

8. И.Я. Олексив, Доказательство существования вложенного решения задачи Плато (1985), Львов. политехн. ин-т, 15с. Деп. в Укр НИИНТИ 19.02.1986г., 655-Ук86.

9. J. Haas, P. Scott, The existence of least area surfaces in 3-manifolds, Trans. AMS 310 (1988), 87–114.

10. R.H. Bing, Approximating surfaces from the side, Ann. of Math. 77 (1963), 145–192.

11. А.А. Тужилин, А.Т. Фоменко, Элементы геометрии и топологии минимальных поверхностей, Наука, М., 1991, 176 с.

12. H. Federer, Hausdorff measure and Lebesgue area, Proc. Nat. Acad. of Sci. USA 37 (1951), no. 2, 90–94.

13. Г. Федерер, Геометрическая теория меры, Наука, М., 1987, 760с.

14. E.J. McShane, Extension of range of functions, Bull. AMS 40 (1934), 837–842.

15. C.D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. 66 (1957), 1–26.

16. T. Rado, On the Problem of Plateau, Berlin: Springer-Verlag, 1933, 109 p.

	On area of 2-spheres in Euclidean space
Author	I.Ya. Oleksiv State University "Lviv Polytechnic", Lviv, Ukraine
Abstract	It is proved that for a 2-sphere embedded in Euclidean 3-space the infimum of the area limits of approximating PL 2-spheres is equal to the Lebesgue area of the 2-sphere.
Keywords	2-sphere, Euclidean 3-space, PL 2-spheres, area limits, Lebesgue area
DOI	doi:10.30970/ms.12.1.15-30
Reference	1. L. Cesari, Surface Area, vol. 35, (Annals of Mathematics Study), Princeton University Press, 1956, 596 pp., pp. 596. 2. R.H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. 65 (1957), 456–483. 3. R.P. Goblirsch, An area for simple surfaces, Ann. of Math. 68 (1958), 231–246. 4. И.Я. Олексив, О площади простой поверхности, ДАН СССР 221 (1975), no. 1, 30–31. 5. А.С. Кронрод, О функциях двух переменных, Успехи матем. наук 5 (1950), no. 1(35), 24–134. 6. K. Kuratowski, Sur les coupures irreductible du plan, Fund. Math. 6 (1924), 130–145. 7. W.H. Meeks, S.-T. Yau, The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglass-Morrey and the analytic proof of Dehn’s lemma, Topology 21 (1982), 409–442. 8. И.Я. Олексив, Доказательство существования вложенного решения задачи Плато (1985), Львов. политехн. ин-т, 15с. Деп. в Укр НИИНТИ 19.02.1986г., 655-Ук86. 9. J. Haas, P. Scott, The existence of least area surfaces in 3-manifolds, Trans. AMS 310 (1988), 87–114. 10. R.H. Bing, Approximating surfaces from the side, Ann. of Math. 77 (1963), 145–192. 11. А.А. Тужилин, А.Т. Фоменко, Элементы геометрии и топологии минимальных поверхностей, Наука, М., 1991, 176 с. 12. H. Federer, Hausdorff measure and Lebesgue area, Proc. Nat. Acad. of Sci. USA 37 (1951), no. 2, 90–94. 13. Г. Федерер, Геометрическая теория меры, Наука, М., 1987, 760с. 14. E.J. McShane, Extension of range of functions, Bull. AMS 40 (1934), 837–842. 15. C.D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. 66 (1957), 1–26. 16. T. Rado, On the Problem of Plateau, Berlin: Springer-Verlag, 1933, 109 p.
Pages	15-30
Volume	12
Issue	1
Year	1999
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On area of 2-spheres in Euclidean space