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A sufficient condition for the convexity of the area of an isoptic curve of an oval. (English) Zbl 1121.52011

Given a plane, closed, convex curve \(C\), one may consider the locus, \(C_{\alpha}\), of those points at which \(C\) subtends an angle of \(\pi-\alpha\) (this is called an isoptic curve). Previous work of W. Cieślak, A. Miernowski and W. Mozgawa [Rend. Semin. Mat. Univ. Padova 96, 37–49 (1996; Zbl 0881.53003)] considered the convexity of \(C_{\alpha}\). Here the author considers the area \(A(\alpha)\) of \(C_{\alpha}\) and discusses the convexity of this function of \(\alpha\). The sufficient conditions given are inequalities for the Fourier coefficients of the support function of \(C\).

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
53C65 Integral geometry
52A10 Convex sets in \(2\) dimensions (including convex curves)

Citations:

Zbl 0881.53003
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References:

[1] W. CIEŚLAK - A. MIERNOWSKI - W. MOZGAWA, Isoptics of a Closed Strictly Convex Curve, Lect. Notes in Math., 1481 (1991), pp. 28-35. Zbl0739.53001 MR1178515 · Zbl 0739.53001
[2] W. CIEŚLAK - A. MIERNOWSKI - W. MOZGAWA, Isoptics of a Closed Strictly Convex Curve II, Rend. Sem. Mat. Univ. Padova, 96 (1996), pp. 37-49. Zbl0881.53003 MR1438287 · Zbl 0881.53003
[3] H. GROEMER, Geometric applications of Fourier series and spherical harmonics, Encyclopedia of Mathematics and its Applications, Cambridge Univ. Press, Cambridge, 1996. Zbl0877.52002 MR1412143 · Zbl 0877.52002
[4] M. A. HURWITZ, Sur quelques applications géométriques des séries de Fourier, Ann. Sci. Ecole Norm. Sup., 19 (1902), pp. 357-408. MR1509016 JFM33.0599.02 · JFM 33.0599.02
[5] A. ZYGMUND, Trigonometric series, Cambridge Univ. Press, vol. I, II, 1968. Zbl0367.42001 MR236587 · Zbl 0367.42001
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