A. Andrei, Y. L. Agrachev, and . Sachkov, Control theory from the geometric viewpoint, of Encyclopaedia of Mathematical Sciences Control Theory and Optimization, II, 2004.

J. A. Andrejewa and R. Klötzler, Zur analytischen L??sung geometrischer Optimierungsaufgaben mittels Dualit??t bei Steuerungsproblemen. Teil I, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.63, issue.1, pp.35-44, 1984.
DOI : 10.1002/zamm.19840640107

J. A. Andrejewa and R. Klötzler, Zur analytischen L??sung geometrischer Optimierungsaufgaben mittels Dualit??t bei Steuerungsproblemen. Teil II, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.12, issue.3, pp.147-153, 1984.
DOI : 10.1002/zamm.19840640303

T. Bayen, Minimization of the area of rotors. En cours de soumission

T. Bayen, E. Oudet, and T. Lachand-robert, Analytic parametrization and volume minimization of three dimensional bodies of constant width
URL : https://hal.archives-ouvertes.fr/hal-00385121

W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Mathematische Annalen, vol.76, issue.4, pp.504-513, 1915.
DOI : 10.1007/BF01458221

T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, 1987.

S. Campi, A. Colesanti, and P. Gronchi, Minimum problems for volumes of convex bodies, Partial differential equations and applications, pp.43-55, 1996.

G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, Convexity and its applications, pp.49-96, 1983.
DOI : 10.1007/978-3-0348-5858-8_3

P. Manfredo and . Do-carmo, Differential geometry of curves and surfaces, N.J, 1976.

J. Dugundji and A. Granas, Fixed point theory. I, of Monografie Matematyczne [Mathematical Monographs]. Pa´nstwowePa´nstwowe Wydawnictwo Naukowe (PWN), 1982.

C. Lawrence, R. F. Evans, and . Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, 1992.

W. J. Firey, Isoperimetric ratios of Reuleaux polygons, Pacific Journal of Mathematics, vol.10, issue.3, pp.823-829, 1960.
DOI : 10.2140/pjm.1960.10.823

J. and C. Fisher, Curves of constant width from a linear viewpoint, Math. Mag, vol.60, issue.3, pp.131-140, 1987.

J. Focke, Symmetrischen-Orbiformen kleinsten Inhalts, Acta Mathematica Academiae Scientiarum Hungaricae, vol.2, issue.1-2, pp.39-68, 1969.
DOI : 10.1007/BF01894567

M. Fujiwara, Analytical proof of Blaschke's theorem on the curve of constant breadth with minimum area, Proc. Tokyo Imp. Acad. Japan, pp.300-302, 1927.

M. Fujiwara, Analytical proof of Blaschke's theorem on the curve of constant bredth with minimum area, Proc. Tokyo Imp. Acad. Japan, pp.300-302, 1931.

M. Fujiwara and S. Kakeya, On some problems of maxima and minima for the curve of constant breadth and the in-resolvable curve of the equilateral triangle, Tohoku Mathemaical Journal, vol.11, pp.92-110, 1917.

M. Ghandehari, An Optimal Control Formulation of the Blaschke???Lebesgue Theorem, Journal of Mathematical Analysis and Applications, vol.200, issue.2, pp.322-331, 1996.
DOI : 10.1006/jmaa.1996.0208

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, vol.224, 1977.

M. Goldberg, Circular-Arc Rotors in Regular Polygons, The American Mathematical Monthly, vol.55, issue.7, pp.393-402, 1948.
DOI : 10.2307/2306128

M. Goldberg, Trammel Rotors in Regular Polygons, The American Mathematical Monthly, vol.64, issue.2, pp.71-78, 1957.
DOI : 10.2307/2310379

M. Goldberg, Rotors in polygons and polyhedra, Mathematics of Computation, vol.14, issue.71, pp.229-239, 1960.
DOI : 10.1090/S0025-5718-1960-0115132-8

P. M. Gruber and J. M. Wills, Handbook of convex geometry, 1993.

M. Evans and I. Harrell, Calculations for convex bodies. example : The rotated Reuleaux triangle

M. Evans and I. Harrell, A direct proof of a theorem of Blaschke and Lebesgue, J. Geom. Anal, vol.12, issue.1, pp.81-88, 2002.

A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics . Birkhäuser Verlag, 2006.

J. Hiriart-urruty and C. Lemaréchal, Fundamentals of convex analysis. Grundlehren Text Editions Abridged version of Convex analysis and minimization algorithms. I, MR1261420 (95m :90001)] and II [ibid. ; MR1295240 (95m :90002)], 1993.

R. Howard, Convex bodies of constant width and constant brightness, Advances in Mathematics, vol.204, issue.1, pp.241-261, 2006.
DOI : 10.1016/j.aim.2005.05.015

R. Klötzler, Beweis einer Vermutung ??bern-Orbiformen kleinsten Inhalts, ZAMM - Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.4, issue.10, pp.557-570, 1975.
DOI : 10.1002/zamm.19750551003

R. Klötzler, Konvexe Bereiche kleinster Oberfl??che bei gegebener Dicke, Zeitschrift f??r Analysis und ihre Anwendungen, vol.4, issue.4, pp.373-383, 1985.
DOI : 10.4171/ZAA/160

A. Kripfganz, Favard???s ???fonction penetrante??? ???a roughly convex function, Optimization, vol.36, issue.4, pp.329-342, 1996.
DOI : 10.1080/01630569108816445

T. Lachand-robert and M. A. Peletier, Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions, Mathematische Nachrichten, vol.173, issue.1, pp.153-176, 2001.
DOI : 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.0.CO;2-2

T. Lachand-robert and E. Oudet, Bodies of constant width in arbitrary dimension, Mathematische Nachrichten, vol.56, issue.7
DOI : 10.1002/mana.200510512

URL : https://hal.archives-ouvertes.fr/hal-00385113

H. Lebesgue, Sur quelques questions de minimum, relatives aux courbes orbiformes, et sur leurs rapports avec le calcul des variations, Journal de Mathématiques, vol.4, issue.8, 1921.

E. B. Lee and L. Markus, Foundations of optimal control theory, 1986.

L. A. Lyusternik, Convex figures and polyhedra. Translated from the Russian by T. Jefferson Smith, 1963.

. Meissner, ¨ Uber die Anwendung von Fourierreihen auf einige Aufgaben der Geometrie und Kinematik, Vierteljahresschr. Naturfor. Ges. Zürich Vierteljahresschr. Naturfor. Ges. Zürich, vol.54, issue.56, pp.309-32942, 1909.

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes. Translated from the Russian by K. N. Trirogoff, 1962.

F. Reuleaux, The kinematics of machinery : Outline of a theory of machines, p.1876

R. Tyrrell-rockafellar, Convex analysis. Princeton Mathematical Series, 1970.

R. Schneider, Convex bodies : the Brunn-Minkowski theory, volume 44 of Encyclopedia of Mathematics and its Applications, 1993.

R. Paul, P. Scott, and . Awyong, Inequalities for convex sets, JIPAM. J. Inequal. Pure Appl. Math, vol.1, issue.6, p.pp, 2000.

K. Toyama, Self-parallel constant mean curvature surfaces. EG-models, www. eg-models. de/ models, 2002.

E. Trélat, Contrôle optimal, Mathématiques Concrètes. [Concrete Mathematics]. Vuibert, Paris, 2005. Théorie & applications. [Theory and applications]

F. A. Valentine, Convex sets. McGraw-Hill Series in Higher Mathematics, 1964.

B. Weissbach, Rotoren im regulären Dreieck, Publ. Math. Debrecen, vol.19, pp.21-27, 1972.

B. Weissbach, Ausbohrung von Rhomben, Wiss. Beitr. Martin-Luther-Univ. Halle- Wittenberg M Beitr. Algebra Geom, vol.8, issue.6, pp.153-158, 1977.

B. Weissbach and . Uber, Orbiformen und ihre Durchmesser, Beiträge Algebra Geom, issue.13, pp.111-125, 1982.

I. M. Yaglom and V. G. Boltyanskii, Convex Figures