A quantitative version of the isoperimetric inequality : the anisotropic case
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 4 (2005) no. 4, p. 619-651
We state and prove a stability result for the anisotropic version of the isoperimetric inequality. Namely if E is a set with small anisotropic isoperimetric deficit, then E is “close” to the Wulff shape set.
Classification:  52A40,  28A75
@article{ASNSP_2005_5_4_4_619_0,
     author = {Esposito, Luca and Fusco, Nicola and Trombetti, Cristina},
     title = {A quantitative version of the isoperimetric inequality : the anisotropic case},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {4},
     year = {2005},
     pages = {619-651},
     zbl = {1170.52300},
     mrnumber = {2207737},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_4_619_0}
}
Esposito, Luca; Fusco, Nicola; Trombetti, Cristina. A quantitative version of the isoperimetric inequality : the anisotropic case. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 4 (2005) no. 4, pp. 619-651. http://www.numdam.org/item/ASNSP_2005_5_4_4_619_0/

[1] M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 91-133. | Numdam | MR 1259102 | Zbl 0842.49016

[2] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford University Press, 2000. | MR 1857292 | Zbl 0957.49001

[3] T. Bonnesen, Über die isoperimetrische Defizit ebener Figuren, Math. Ann. 91 (1924), 252-268. | JFM 50.0487.03 | MR 1512192

[4] Y. D. Burago and V. A. Zalgaller, “Geometric Inequalities”, Grund. Math. Wissen., Springer, 1988. | MR 936419 | Zbl 0633.53002

[5] G. Buttazzo, V. Ferone and B. Kawohl, Minimum problems over sets of concave functions and related questions, Math. Nachr. 173 (1995), 71-89. | MR 1336954 | Zbl 0835.49001

[6] M. Chlebík, A. Cianchi and N. Fusco, The perimeter inequality for Steiner symmatrization: cases of equality, Ann. of Math. 162 (2005), 525-555. | MR 2178968 | Zbl 1087.28003

[7] B. Dacorogna and C. E. Pfister, Wulff theorem and best constant in Sobolev inequality, J. Math. Pures Appl. 71 (1992), 97-118. | MR 1170247 | Zbl 0676.46031

[8] E. De Giorgi, Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8) 5 (1958), 33-44. | MR 98331 | Zbl 0116.07901

[9] A. Dinghas, Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen, Z. Krist. 105 (1944), 304-314. | MR 12454 | Zbl 0028.43001

[10] R. M. Dudley, Metric entropy of some classes of sets with differentiable boundaries, J. Approx. Theory 10 (1974), 227-236. | MR 358168 | Zbl 0275.41011

[11] I. Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. London 432 (1991), 125-145. | MR 1116536 | Zbl 0725.49017

[12] I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh 119A (1991), 125-136. | MR 1130601 | Zbl 0752.49019

[13] B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in n , Trans. Amer. Math. Soc. 314 (1989), 619-638. | MR 942426 | Zbl 0679.52007

[14] P. M. Gruber, Aspects of Approximation of Convex Bodies, In: “Handbook of Convex Geometry”, P. M. Gruber and J. M. Wills (eds.), Elsevier, 1993, 319-345. | MR 1242984 | Zbl 0791.52007

[15] R. R. Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428 (1992), 161-176. | MR 1166511 | Zbl 0746.52012

[16] R. R. Hall, W. K. Hayman and A. W. Weitsman, On asymmetry and capacity, J. Anal. Math. 56 (1991), 87-123. | MR 1243100 | Zbl 0747.31004

[17] A. Hertle, On the problem of well-posedness for the Radon transform, In: “Mathematical Aspects of Computerized Tomography”, Proc. Oberwolfach 1980, Lect. Notes Medic. Inform., Vol. 8, 1981, 36-44. | MR 720394 | Zbl 0555.46020

[18] M. Longinetti, Some questions of stability in the reconstruction of plane convex bodies from projections, Inverse Problems 1 (1985), 87-97. | MR 787854 | Zbl 0597.52001

[19] J. Taylor, Existence and structure of solutions to a class of nonelliptic variational problems, Sympos. Math. 14 (1974), 499-508. | MR 420407 | Zbl 0317.49053

[20] J. Taylor, Unique structure of solutions to a class of nonelliptic variational problems, Proc. Sympos. Pure Math. 27 (1975), 419-427. | MR 388225 | Zbl 0317.49054

[21] J. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), 568-588. | MR 493671 | Zbl 0392.49022

[22] G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallfläschen, Z. Krist. 34 (1901), 449-530.