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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

Bernd Kawohl , and  Guido Sweers

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.

Keywords: Construction and approximation, convex sets of constant width, formula of Euler.
Mathematics Subject Classification: Primary: 52A10; Secondary: 52A27, 46B28.
Citation: Bernd Kawohl, Guido Sweers. On a formula for sets of constant width in 2d. Communications on Pure and 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.,

[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.

[3]

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

[4]

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

[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.,

[6]

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

[7]

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

[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.,

[9]

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

[10]

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

[11]

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

[12]

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

[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).,

[14]

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

[15]

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

[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.

[17]

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

[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. 

[19]

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

[20]

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

[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.

[22]

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

[23]

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

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.,

[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.

[3]

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

[4]

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

[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.,

[6]

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

[7]

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

[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.,

[9]

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

[10]

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

[11]

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

[12]

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

[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).,

[14]

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

[15]

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

[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.

[17]

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

[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. 

[19]

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

[20]

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

[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.

[22]

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

[23]

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

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
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Figure 2.  The relation between the directional width and the support function
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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 } $
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Figure 4.  $ G $ and $ G_{\varepsilon } $ with the auxiliary circles
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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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