## Isoperimetric inequalities in crystallography

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- by Antonio Ros
- J. Amer. Math. Soc.
**17**(2004), 373-388 - DOI: https://doi.org/10.1090/S0894-0347-03-00447-8
- Published electronically: December 2, 2003
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## Abstract:

Given a cubic space group $\mathcal G$ (viewed as a finite group of isometries of the torus $T=\mathbb {R}^3/\mathbb {Z}^3$), we obtain sharp isoperimetric inequalities for $\mathcal G$-invariant regions. These inequalities depend on the minimum number of points in an orbit of $\mathcal G$ and on the largest Euler characteristic among nonspherical $\mathcal G$-symmetric surfaces minimizing the area under volume constraint (we also give explicit estimates of this second invariant for the various crystallographic cubic groups $\mathcal G$). As an example, we prove that any surface dividing $T$ into two equal volumes with the same (orientation-preserving) symmetries as the A. Schoen minimal Gyroid has area at least $3.00$ (the conjectured minimizing surface in this case is the Gyroid itself whose area is $3.09$).## References

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## Bibliographic Information

**Antonio Ros**- Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
- Email: aros@ugr.es
- Received by editor(s): March 17, 2003
- Published electronically: December 2, 2003
- Additional Notes: Partially supported by MCYT-FEDER research projects BFM2001-3318
- © Copyright 2003 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**17**(2004), 373-388 - MSC (2000): Primary 53A10, 53C42, 20H15
- DOI: https://doi.org/10.1090/S0894-0347-03-00447-8
- MathSciNet review: 2051615