American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids
  • CPAA Home
  • This Issue
  • Next Article
    On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction
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 & 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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The relation between the directional width and the support function
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 4.  $ G $ and $ G_{\varepsilon } $ with the auxiliary circles
Figure Options

Download full-size image

Download as PowerPoint slide

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 $j_{1}\left( \theta \right) = \int_{\beta -\theta \left( \beta -\alpha \right) }^{\beta }e^{is}ds$ .
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Roland Hildebrand. Barriers on projective convex sets. Conference Publications, 2011, 2011 (Special) : 672-683. doi: 10.3934/proc.2011.2011.672
[2]

Jacek Banasiak, Aleksandra Puchalska. Generalized network transport and Euler-Hille formula. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1873-1893. doi: 10.3934/dcdsb.2018185
[3]

José Natário. An elementary derivation of the Montgomery phase formula for the Euler top. Journal of Geometric Mechanics, 2010, 2 (1) : 113-118. doi: 10.3934/jgm.2010.2.113
[4]

Juan Pablo Rincón-Zapatero. Hopf-Lax formula for variational problems with non-constant discount. Journal of Geometric Mechanics, 2009, 1 (3) : 357-367. doi: 10.3934/jgm.2009.1.357
[5]

Kais Hamza, Fima C. Klebaner. On nonexistence of non-constant volatility in the Black-Scholes formula. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 829-834. doi: 10.3934/dcdsb.2006.6.829
[6]

Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
[7]

Lisa Hernandez Lucas. Properties of sets of subspaces with constant intersection dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052
[8]

Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103
[9]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292
[10]

Joel Spruck, Ling Xiao. Convex spacelike hypersurfaces of constant curvature in de Sitter space. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2225-2242. doi: 10.3934/dcdsb.2012.17.2225
[11]

Jean-François Babadjian, Clément Mifsud, Nicolas Seguin. Relaxation approximation of Friedrichs' systems under convex constraints. Networks & Heterogeneous Media, 2016, 11 (2) : 223-237. doi: 10.3934/nhm.2016.11.223
[12]

Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim, Dae-Woon Lim. New construction methods of quaternary periodic complementary sequence sets. Advances in Mathematics of Communications, 2010, 4 (1) : 61-68. doi: 10.3934/amc.2010.4.61
[13]

Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519
[14]

Jutamas Kerdkaew, Rabian Wangkeeree. Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2651-2673. doi: 10.3934/jimo.2019074
[15]

Sun-Yung Alice Chang, Xi-Nan Ma, Paul Yang. Principal curvature estimates for the convex level sets of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1151-1164. doi: 10.3934/dcds.2010.28.1151
[16]

Sebastián Buedo-Fernández. Global attraction in a system of delay differential equations via compact and convex sets. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3171-3181. doi: 10.3934/dcdsb.2020056
[17]

R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39
[18]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

[19]

Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[20]

Eric Baer, Alessio Figalli. Characterization of isoperimetric sets inside almost-convex cones. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 1-14. doi: 10.3934/dcds.2017001

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (210)
  • HTML views (225)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy