• 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

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

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

[2]

Sascha Kurz. The interplay of different metrics for the construction of constant dimension codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021069

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Xianhong Xie, Yi Ouyang, Honggang Hu, Ming Mao. Construction of three classes of strictly optimal frequency-hopping sequence sets. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022024

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (233)
  • HTML views (226)
  • Cited by (0)

Other articles
by authors

[Back to Top]