Approximation of convex bodies by axially symmetric bodies
Author:
Marek Lassak
Journal:
Proc. Amer. Math. Soc. 130 (2002), 3075-3084
MSC (1991):
Primary 52A10, 52A27
DOI:
https://doi.org/10.1090/S0002-9939-02-06404-3
Published electronically:
March 14, 2002
Erratum:
Proc. Amer. Math. Soc. 131 (2003), 2301-2301.
MathSciNet review:
1908932
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Abstract: Let $C$ be an arbitrary planar convex body. We prove that $C$ contains an axially symmetric convex body of area at least $\frac {2}{3}|C|$. Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least $\frac {1}{2}|C|$ in $C$, and we can circumscribe a homothetic rhombus of area at most $2|C|$ about $C$. The homothety ratio is at most $2$. Those factors $\frac {1}{2}$ and $2$, as well as the ratio $2$, cannot be improved.
- A. S. Besicovitch, Measure of a asymmetry of convex curves, J. London. Math. Soc. 23 (1948), 237-240.
- Andrzej B. Buda and Kurt Mislow, On a measure of axiality for triangular domains, Elem. Math. 46 (1991), no. 3, 65–73. MR 1113766
- G. D. Chakerian and S. K. Stein, On measures of symmetry of convex bodies, Canadian J. Math. 17 (1965), 497–504. MR 177349, DOI https://doi.org/10.4153/CJM-1965-049-3
- C. M. Christensen, Kvadrat indskrevet i convex figur, Mat. Tidsskr. B (1950), 22-26.
- B. Abel de Valcourt, Measures of axial symmetry for ovals, Bull. Amer. Math. Soc. 72 (1966), 289–290. MR 187142, DOI https://doi.org/10.1090/S0002-9904-1966-11495-7
- B. Abel de Valcourt, Measures of axial symmetry for ovals, Israel J. Math. 4 (1966), 65–82. MR 203589, DOI https://doi.org/10.1007/BF02937452
- B. Abel de Valcourt, Axially symmetric polygons inscribed in and circumscribed about convex sets, Elem. Math. 22 (1967), 121–133. MR 226504
- E. G. Eggleston, Convexity, Cambridge University Press, Cambridge, 1969. (1st ed. 1958, )
- A. Emch, Some properties of closed convex curves in a plane, Amer. J. Math. 35 (1913), 407-412.
- Rudolf Fleischer, Kurt Mehlhorn, Günter Rote, Emo Welzl, and Chee Yap, Simultaneous inner and outer approximation of shapes, Algorithmica 8 (1992), no. 5-6, 365–389. 1990 Computational Geometry Symposium (Berkeley, CA, 1990). MR 1195158, DOI https://doi.org/10.1007/BF01758852
- 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.
- Branko Grünbaum, Measures of symmetry for convex sets, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 233–270. MR 0156259
- J. L. Hodges, Jr., An extremal problem of geometry, J. London Math. Soc. 26 (1951), 312-313.
- F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume 1948, pp. 187-204.
- F. Krakowski, Bemerkung zur einer Arbeit von W. Nohl, Elem. Math. 18 (1963), 60-61.
- Marek Lassak, Approximation of convex bodies by triangles, Proc. Amer. Math. Soc. 115 (1992), no. 1, 207–210. MR 1057956, DOI https://doi.org/10.1090/S0002-9939-1992-1057956-1
- Marek Lassak, Approximation of convex bodies by rectangles, Geom. Dedicata 47 (1993), no. 1, 111–117. MR 1230108, DOI https://doi.org/10.1007/BF01263495
- W. Nohl, Die innere axiale Symmetrie zentrischer Eibereiche der euklidischen Ebene, Elem. Math. 17 (1962), 59-63.
- Otfried Schwarzkopf, Ulrich Fuchs, Günter Rote, and Emo Welzl, Approximation of convex figures by pairs of rectangles, STACS 90 (Rouen, 1990) Lecture Notes in Comput. Sci., vol. 415, Springer, Berlin, 1990, pp. 240–249. MR 1063318, DOI https://doi.org/10.1007/3-540-52282-4_47
- Otfried Schwarzkopf, Ulrich Fuchs, Günter Rote, and Emo Welzl, Approximation of convex figures by pairs of rectangles, Comput. Geom. 10 (1998), no. 2, 77–87. MR 1614621, DOI https://doi.org/10.1016/S0925-7721%2896%2900019-3
- K. Zindler, Über konvexe Gebilde II, Monatsh. Math. 31 (1921), 25-56.
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Additional Information
Marek Lassak
Affiliation:
Instytut Matematyki i Fizyki ATR, 85-796 Bydgoszcz, Poland
Address at time of publication:
Institut für Informatik, FU Berlin, D-14195, Berlin, Germany
Email:
lassak@mail.atr.bydgoszcz.pl
Keywords:
Convex body,
axial symmetry,
rhombus,
isosceles triangle,
area,
approximation
Received by editor(s):
March 1, 2000
Received by editor(s) in revised form:
May 1, 2001
Published electronically:
March 14, 2002
Additional Notes:
This research was supported by Deutsche Forschungsgemeinschaft
Communicated by:
Wolfgang Ziller
Article copyright:
© Copyright 2002
American Mathematical Society