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Approximation of convex bodies by axially symmetric bodies

Author(s): Marek Lassak
Journal: Proc. Amer. Math. Soc. 130 (2002), 3075-3084.
MSC (1991): Primary 52A10, 52A27
Posted: March 14, 2002
Errata: Proc. Amer. Math. Soc. 131 (2003), 2301.
MathSciNet review: 1908932
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Abstract: Let be an arbitrary planar convex body. We prove that contains an axially symmetric convex body of area at least $\frac{2}{3}\vert C\vert$ . Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least $\frac{1}{2}\vert C\vert$ in , and we can circumscribe a homothetic rhombus of area at most $2\vert C\vert$ about . The homothety ratio is at most . Those factors $\frac{1}{2}$ and , as well as the ratio , cannot be improved.

References:

1.: A. S. Besicovitch, Measure of a asymmetry of convex curves, J. London. Math. Soc. 23 (1948), 237-240. MR 10:320r
2.: A. B. Buda and K. Mislow, On measure of axiality for triangular domains, Elem. Math. 46 (1991), 65-73. MR 92d:52017
3.: G. D. Chakerian and S. K. Stein, Measures of symmetry of convex bodies, Canad. J. Math. 17 (1965), 497-504. MR 31:1612
4.: C. M. Christensen, Kvadrat indskrevet i convex figur, Mat. Tidsskr. B (1950), 22-26. MR 12:525f
5.: B. A. deValcourt, Measures of axial symmetry for ovals, Bull. Amer. Math. Soc. 72 (1966), 289-290. MR 32:4596
6.: B. A. deValcourt, Measures of axial symmetry for ovals, Isr. J. Math. 4 (1966), 65-82. MR 34:3439
7.: B. A. deValcourt, Axially symmetric polygons inscribed in and circumscribed about convex sets, Elem. Math. 22 (1967), 121-133. MR 37:2093
8.: E. G. Eggleston, Convexity, Cambridge University Press, Cambridge, 1969. (1st ed. 1958, MR 23:A2123)
9.: A. Emch, Some properties of closed convex curves in a plane, Amer. J. Math. 35 (1913), 407-412.
10.: R. Fleischer, K. Mehlhorn, G. Rote, E. Welzl, and C. Yap, Simultaneous inner and outer approximation of shapes, Algorithmica 8, 1992, 365-389. MR 93j:68213
11.: W. Gross, Über affine Geometrie XIII: Eine Minimumeigenschaft der Ellipse und des Elipsoids, Ber. Ver. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl. 70 (1918), 38-54.
12.: B. Grünbaum, Measures of symmetry of convex sets, Convexity (Seattle 1961), in Proc. Sympos. Pure Math. 7, Amer. Math. Soc., Providence, R.I., 1963, pp. 233-270. MR 27:6187
13.: J. L. Hodges, Jr., An extremal problem of geometry, J. London Math. Soc. 26 (1951), 312-313. MR 14:495f
14.: F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume 1948, pp. 187-204. MR 10:719b
15.: F. Krakowski, Bemerkung zur einer Arbeit von W. Nohl, Elem. Math. 18 (1963), 60-61.
16.: M. Lassak, Approximation of convex bodies by triangles, Proc. Amer. Math. Soc. 115 (1992), 207-210. MR 92h:52004
17.: M. Lassak, Approximation of convex bodies by rectangles, Geom. Dedicata 47 (1993), 111-117. MR 94f:52004
18.: W. Nohl, Die innere axiale Symmetrie zentrischer Eibereiche der euklidischen Ebene, Elem. Math. 17 (1962), 59-63.
19.: O. Schwarzkopf, U. Fuchs, G. Rote and E. Welzl, Approximation of convex figures by pairs of rectangles, Proc. 7-th Sympos. Teoret. Aspects Comput. Sci., In STAC 90, pp. 240-249, Lecture Notes in Comput. Sci. 415, Springer, Berlin. MR 91e:68148
20.: O. Schwarzkopf, U. Fuchs, G. Rote and E. Welzl, Approximation of convex figures by pairs of rectangles, Comp. Geom. Theor. Appl. 10 (1998), 77-87. MR 99a:52019
21.: K. Zindler, Über konvexe Gebilde II, Monatsh. Math. 31 (1921), 25-56.

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