Measures of $N$-fold symmetry for convex sets
Authors:
Charles K. Chui and Milton N. Parnes
Journal:
Proc. Amer. Math. Soc. 26 (1970), 480-486
MSC:
Primary 52.30
DOI:
https://doi.org/10.1090/S0002-9939-1970-0264514-0
MathSciNet review:
0264514
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Abstract | References | Similar Articles | Additional Information
Abstract: If a convex set $S$ is $3$-fold symmetric about a point $0 \in S$, then any $3$-star contained in $S$ with vertex 0 is no smaller than any other parallel $3$-star contained in $S$. In this paper, among other results, we establish the converse. Consequently, we find two measures of $n$-fold symmetry, one for $n = 2,3$ and the other for each $n \geqq 2$.
- H. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. MR 0124813
- 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
- P. C. Hammer, Diameters of convex bodies, Proc. Amer. Math. Soc. 5 (1954), 304–306. MR 61398, DOI https://doi.org/10.1090/S0002-9939-1954-0061398-1 M. Parnes, Symmetrization and conformal mapping, Ph.D. thesis, Wayne State Univ., Detroit, Mich., 1968.
- Milton N. Parnes, A distortion theorem for doubly connected regions, Proc. Amer. Math. Soc. 26 (1970), 85–91. MR 265569, DOI https://doi.org/10.1090/S0002-9939-1970-0265569-X
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Additional Information
Keywords:
<IMG WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$N$">-fold symmetry,
convex sets,
similarity invariant measure,
<IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img15.gif" ALT="$n$">-maximal property,
<IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$n$">-supporting-line property,
<IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$n$">-star
Article copyright:
© Copyright 1970
American Mathematical Society