A quantitative version of the isoperimetric inequality : the anisotropic case

 Access to full text

 Full (PDF)

1. [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. Zbl0842.49016MR1259102
2. [2] L. Ambrosio, N. Fusco and D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford University Press, 2000. Zbl0957.49001MR1857292
3. [3] T. Bonnesen, Über die isoperimetrische Defizit ebener Figuren, Math. Ann. 91 (1924), 252–268. Zbl50.0487.03MR1512192JFM50.0487.03
4. [4] Y. D. Burago and V. A. Zalgaller, “Geometric Inequalities”, Grund. Math. Wissen., Springer, 1988. Zbl0633.53002MR936419
5. [5] G. Buttazzo, V. Ferone and B. Kawohl, Minimum problems over sets of concave functions and related questions, Math. Nachr. 173 (1995), 71–89. Zbl0835.49001MR1336954
6. [6] M. Chlebík, A. Cianchi and N. Fusco, The perimeter inequality for Steiner symmatrization: cases of equality, Ann. of Math. 162 (2005), 525–555. Zbl1087.28003MR2178968
7. [7] B. Dacorogna and C. E. Pfister, Wulff theorem and best constant in Sobolev inequality, J. Math. Pures Appl. 71 (1992), 97–118. Zbl0676.46031MR1170247
8. [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. Zbl0116.07901MR98331
9. [9] A. Dinghas, Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen, Z. Krist. 105 (1944), 304–314. Zbl0028.43001MR12454
10. [10] R. M. Dudley, Metric entropy of some classes of sets with differentiable boundaries, J. Approx. Theory 10 (1974), 227–236. Zbl0275.41011MR358168
11. [11] I. Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. London 432 (1991), 125–145. Zbl0725.49017MR1116536
12. [12] I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh 119A (1991), 125–136. Zbl0752.49019MR1130601
13. [13] B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in ${ℝ}^n$, Trans. Amer. Math. Soc. 314 (1989), 619–638. Zbl0679.52007MR942426
14. [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. Zbl0791.52007MR1242984
15. [15] R. R. Hall, A quantitative isoperimetric inequality in $n$-dimensional space, J. Reine Angew. Math. 428 (1992), 161–176. Zbl0746.52012MR1166511
16. [16] R. R. Hall, W. K. Hayman and A. W. Weitsman, On asymmetry and capacity, J. Anal. Math. 56 (1991), 87–123. Zbl0747.31004MR1243100
17. [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. Zbl0555.46020MR720394
18. [18] M. Longinetti, Some questions of stability in the reconstruction of plane convex bodies from projections, Inverse Problems 1 (1985), 87–97. Zbl0597.52001MR787854
19. [19] J. Taylor, Existence and structure of solutions to a class of nonelliptic variational problems, Sympos. Math. 14 (1974), 499–508. Zbl0317.49053MR420407
20. [20] J. Taylor, Unique structure of solutions to a class of nonelliptic variational problems, Proc. Sympos. Pure Math. 27 (1975), 419–427. Zbl0317.49054MR388225
21. [21] J. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), 568–588. Zbl0392.49022MR493671
22. [22] G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallfläschen, Z. Krist. 34 (1901), 449–530. 

1. Carlo Nitsch, The Quantitative Isoperimetric Inequality for Planar Convex Domains
2. Stefano Amato, Giovanni Bellettini, Lucia Tealdi, Anisotropic mean curvature on facets and relations with capillarity