TY - JOUR
T1 - Contraction of convex hypersurfaces by their affine normal
AU - Andrews, Ben
PY - 1996
Y1 - 1996
N2 - An affine-invariant evolution equation for convex hypersurfaces in Euclidean space is defined by assigning to each point a velocity equal to the affine normal vector. For an arbitrary compact, smooth, strictly convex initial hypersurface, it is shown that this deformation produces a unique, smooth family of convex hypersurfaces, which converge to a point in finite time. Furthermore, the hypersurfaces converge smoothly to an ellipsoid after rescaling about the final point to make the enclosed volume constant. The result leads to simple proofs of some affine-geometric isoperimetric inequalities.
AB - An affine-invariant evolution equation for convex hypersurfaces in Euclidean space is defined by assigning to each point a velocity equal to the affine normal vector. For an arbitrary compact, smooth, strictly convex initial hypersurface, it is shown that this deformation produces a unique, smooth family of convex hypersurfaces, which converge to a point in finite time. Furthermore, the hypersurfaces converge smoothly to an ellipsoid after rescaling about the final point to make the enclosed volume constant. The result leads to simple proofs of some affine-geometric isoperimetric inequalities.
UR - http://www.scopus.com/inward/record.url?scp=0030092746&partnerID=8YFLogxK
U2 - 10.4310/jdg/1214458106
DO - 10.4310/jdg/1214458106
M3 - Article
AN - SCOPUS:0030092746
SN - 0022-040X
VL - 43
SP - 207
EP - 230
JO - Journal of Differential Geometry
JF - Journal of Differential Geometry
IS - 2
ER -