View source for Convex integration

One of the methods developed by M. Gromov to prove the [[H-principle|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202301.png" />-principle]]. The essence of this method is contained in the following statement: If the convex hull of some path-connected subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202302.png" /> contains a small neighbourhood of the origin, then there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202303.png" /> whose derivative sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202304.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202305.png" />. This is equivalent to saying that the differential relation for mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202306.png" /> given by requiring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202307.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202308.png" /> satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c1202309.png" />-principle. More generally, the method of convex integration allows one to prove the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023010.png" />-principle for so-called ample relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023011.png" />. In the simplest case of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023012.png" />-jet bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023013.png" /> over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023014.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023015.png" />, this means that the convex hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023016.png" /> is all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023017.png" /> for any fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023019.png" /> (notice that this fibre is an affine space). The extension to arbitrary dimension and higher-order jet bundles is achieved by studying codimension-one hyperplane fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023021.png" /> and intermediate affine bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023022.png" /> defined in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120230/c12023023.png" />.

One particular application of convex integration is to the construction of divergence-free vector fields and related geometric problems.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,   "Partial differential relations" , ''Ergebn. Math. Grenzgeb. (3)'' , '''9''' , Springer  (1986)  {{MR|0864505}} {{ZBL|0651.53001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Spring,   "Convex integration theory" , ''Monogr. Math.'' , '''92''' , Birkhäuser  (1998)  {{MR|1488424}} {{ZBL|0997.57500}} </TD></TR></table>