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July  2019, 18(4): 2117-2131. doi: 10.3934/cpaa.2019095

On a formula for sets of constant width in 2d

Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany

* Corresponding author

Received  August 2018 Revised  August 2018 Published  January 2019

A formula for smooth orbiforms originating from Euler can be adjusted to describe all sets of constant width in 2d. Moreover, the formula allows short proofs of some laborious approximation results for sets of constant width.

Citation: Bernd Kawohl, Guido Sweers. On a formula for sets of constant width in 2d. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2117-2131. doi: 10.3934/cpaa.2019095
References:
[1]

A. D. Alexandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Ann. Uchenye Zapiski Math., Ser. 6, (1939), 3–35., Google Scholar

[2]

E. Barbier, Note sur le problème de l'aiguille et le jeu du joint couvert, J. Math. Pures Appl., 2e série, tome 5 (1860), 273–286.Google Scholar

[3]

W. Blaschke, Einige Bemerkungen über Kurven und Flächen von konstanter Breite, Ber. Verh. Sächs. Akad. Leipzig, 67 (1915), 290-297. Google Scholar

[4]

W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Mathem. Annalen, 176 (1915), 504-513. doi: 10.1007/BF01458221. Google Scholar

[5]

J. Böhm, Convex bodies of constant width, in Mathematical Models from the Collections of Universities and Museums (ed. G. Fischer), Vieweg, Braunschweig, (1986), 49–56., Google Scholar

[6]

J. Böhm and E. Quaisser, Schönheit und Harmonie Geometrischer Formen–Sphäroformen und Symmetrische Körper, Akademie Verlag, Berlin, 1991., Google Scholar

[7]

G. D. Chakerian, Sets of Constant Width, Pacific J. Math., 19 (1966), 13-21. Google Scholar

[8]

G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, in Convexity and its Applications (eds. P. M. Gruber and J. M. Wills), Birkhauser, (1983), 49–96., Google Scholar

[9]

L. Euler, De curvis triangularibus, Acta Academiae Scientarum Imperialis Petropolitinae, 1778 (1781), 3-30 (Opera Omnia: Series 1, Volume 28, pp. 298-321).Google Scholar

[10]

P. C. Hammer and A. Sobczyk, Planar line families. Ⅰ, Proc. Amer. Math. Soc., 4 (1953), 226-233. doi: 10.2307/2031796. Google Scholar

[11]

P. C. Hammer and A. Sobczyk, Planar line families. Ⅱ, Proc. Amer. Math. Soc., 4 (1953), 341-349. doi: 10.2307/2032127. Google Scholar

[12]

P. C. Hammer, Constant breadth curves in the plane, Proc. Amer.Math. Soc., 6 (1955), 333-334. doi: 10.2307/2032370. Google Scholar

[13]

D. Hilbert and St. Cohn-Vossen, Geometry and The Imagination, AMS Chelsea, Providence, R.I., 1952 (transl. from the German: Anschauliche Geometrie, Springer, Berlin, 1932)., Google Scholar

[14]

I. M. Jaglom and W. G. Boltjanski, Konvexe Figuren, VEB Deutscher Verlag der Wissenschaften, 1956., Google Scholar

[15]

B. Kawohl and Ch. Weber, Meissner's mysterious bodies, The Mathematical Intelligencer, 33 (2011), 94-101. doi: 10.1007/s00283-011-9239-y. Google Scholar

[16]

T. Lachand-Robert and É. Oudet, Bodies of constant width in arbitrary dimension, Mathe-matische Nachrichten, 280 (2007), 740-750. doi: 10.1002/mana.200510512. Google Scholar

[17]

F. Malagoli, An optimal control theory approach to the Blaschke-Lebesgue theorem, J. Convex Anal., 16 (2009), 391-407. Google Scholar

[18]

E. Meissner, Über die Anwendung der Fourier-Reihen auf einige Aufgaben der geometrie und kinematik, Vierteljahrsschr. Nat.forsch. Ges. Zür, 54 (1909), 309-329. Google Scholar

[19]

E. Meissner, Über Punktmengen konstanter Breite, Vierteljahrsschr. Nat.forsch. Ges. Zür, 56 (1911), 42-50. Google Scholar

[20]

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren. Proben mathematischen Denkens für Liebhaber der Mathematik, 2. Aufl. Julius Springer, Berlin, 1933., Google Scholar

[21]

Sh. Tanno, $C^{\infty }$-approximation of continuous ovals of constant width, J. Math. Soc. Japan, 28 (1976), 384-395. doi: 10.2969/jmsj/02820384. Google Scholar

[22]

B. Wegner, Analytic approximation of continuous ovals of constant width, J. Math. Soc. Japan, 29 (1977), 537-540. doi: 10.2969/jmsj/02930537. Google Scholar

[23]

Mathematical Etudes Foundation, Russia. Available from http://www.etudes.ru/en/etudes/reuleaux-triangle/Google Scholar

show all references

References:
[1]

A. D. Alexandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Ann. Uchenye Zapiski Math., Ser. 6, (1939), 3–35., Google Scholar

[2]

E. Barbier, Note sur le problème de l'aiguille et le jeu du joint couvert, J. Math. Pures Appl., 2e série, tome 5 (1860), 273–286.Google Scholar

[3]

W. Blaschke, Einige Bemerkungen über Kurven und Flächen von konstanter Breite, Ber. Verh. Sächs. Akad. Leipzig, 67 (1915), 290-297. Google Scholar

[4]

W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Mathem. Annalen, 176 (1915), 504-513. doi: 10.1007/BF01458221. Google Scholar

[5]

J. Böhm, Convex bodies of constant width, in Mathematical Models from the Collections of Universities and Museums (ed. G. Fischer), Vieweg, Braunschweig, (1986), 49–56., Google Scholar

[6]

J. Böhm and E. Quaisser, Schönheit und Harmonie Geometrischer Formen–Sphäroformen und Symmetrische Körper, Akademie Verlag, Berlin, 1991., Google Scholar

[7]

G. D. Chakerian, Sets of Constant Width, Pacific J. Math., 19 (1966), 13-21. Google Scholar

[8]

G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, in Convexity and its Applications (eds. P. M. Gruber and J. M. Wills), Birkhauser, (1983), 49–96., Google Scholar

[9]

L. Euler, De curvis triangularibus, Acta Academiae Scientarum Imperialis Petropolitinae, 1778 (1781), 3-30 (Opera Omnia: Series 1, Volume 28, pp. 298-321).Google Scholar

[10]

P. C. Hammer and A. Sobczyk, Planar line families. Ⅰ, Proc. Amer. Math. Soc., 4 (1953), 226-233. doi: 10.2307/2031796. Google Scholar

[11]

P. C. Hammer and A. Sobczyk, Planar line families. Ⅱ, Proc. Amer. Math. Soc., 4 (1953), 341-349. doi: 10.2307/2032127. Google Scholar

[12]

P. C. Hammer, Constant breadth curves in the plane, Proc. Amer.Math. Soc., 6 (1955), 333-334. doi: 10.2307/2032370. Google Scholar

[13]

D. Hilbert and St. Cohn-Vossen, Geometry and The Imagination, AMS Chelsea, Providence, R.I., 1952 (transl. from the German: Anschauliche Geometrie, Springer, Berlin, 1932)., Google Scholar

[14]

I. M. Jaglom and W. G. Boltjanski, Konvexe Figuren, VEB Deutscher Verlag der Wissenschaften, 1956., Google Scholar

[15]

B. Kawohl and Ch. Weber, Meissner's mysterious bodies, The Mathematical Intelligencer, 33 (2011), 94-101. doi: 10.1007/s00283-011-9239-y. Google Scholar

[16]

T. Lachand-Robert and É. Oudet, Bodies of constant width in arbitrary dimension, Mathe-matische Nachrichten, 280 (2007), 740-750. doi: 10.1002/mana.200510512. Google Scholar

[17]

F. Malagoli, An optimal control theory approach to the Blaschke-Lebesgue theorem, J. Convex Anal., 16 (2009), 391-407. Google Scholar

[18]

E. Meissner, Über die Anwendung der Fourier-Reihen auf einige Aufgaben der geometrie und kinematik, Vierteljahrsschr. Nat.forsch. Ges. Zür, 54 (1909), 309-329. Google Scholar

[19]

E. Meissner, Über Punktmengen konstanter Breite, Vierteljahrsschr. Nat.forsch. Ges. Zür, 56 (1911), 42-50. Google Scholar

[20]

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren. Proben mathematischen Denkens für Liebhaber der Mathematik, 2. Aufl. Julius Springer, Berlin, 1933., Google Scholar

[21]

Sh. Tanno, $C^{\infty }$-approximation of continuous ovals of constant width, J. Math. Soc. Japan, 28 (1976), 384-395. doi: 10.2969/jmsj/02820384. Google Scholar

[22]

B. Wegner, Analytic approximation of continuous ovals of constant width, J. Math. Soc. Japan, 29 (1977), 537-540. doi: 10.2969/jmsj/02930537. Google Scholar

[23]

Mathematical Etudes Foundation, Russia. Available from http://www.etudes.ru/en/etudes/reuleaux-triangle/Google Scholar

Figure 1.  Two sets of constant width with a boundary of circular arcs. On the left the Reuleaux triangle. The segments connect boundary points with opposite normal directions for which the constant width is attained. The points denote the centers of rotation
Figure 2.  The relation between the directional width and the support function
Figure 3.  Graphs of the functions $ a $ in Recipe 3.1 with the corresponding sets of constant width for $ r = \left\Vert a\right\Vert _{\infty } $
Figure 4.  $ G $ and $ G_{\varepsilon } $ with the auxiliary circles
Figure 5.  The dark (red) part contains the values $\int_\alpha ^\beta {\rho \left( s \right)} {e^{is}}ds$ for all allowed $\rho $ . The bounding curves $j_{0}(\cdot )$ , $j_{1}(\cdot )$ correspond to the extreme cases in (33) as function of $ \theta $ : $j_{0}\left( \theta \right) =\int_ \alpha ^{\alpha +\theta \left( \beta - \alpha \right) }e^{is}ds$ and $j_{1}\left( \theta \right) = \int_{\beta -\theta \left( \beta -\alpha \right) }^{\beta }e^{is}ds$ .
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