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## An application of curvature flows to convex geometry

### Alina Stancu

University of Massachusetts Lowell

###
Thursday, October 20, 2005

L01 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** Curvature flows are characterized by the
deformation of geometric objects such as metrics, mappings, or
submanifolds by geometric quantities depending on curvature. They can
be used to understand some canonical or optimal states which are the
states preferred by the evolving geometric object. For example, closed
convex hypersurfaces in *R^{n+1}* evolving under the affine
curvature flow __tend__ to become ellipsoids. We will use this flow
to answer, under some assumptions, a seemingly unrelated question
known as the floating body problem.\par Let *K \subset
R^{n+1}* be a convex body and let *\delta >0* a
sufficiently small real number. We call the floating body associated
to *K*, denoted *K_{\delta}*, the intersection of all half-spaces in
*R^{n+1}* whose complements cut from *K* a cap of volume
*\delta*. Conjectured in the *90*'s, the floating body problem states
that *K* is homothetic to *K_{\delta}* if and only if *K* is an
ellipsoid.

This talk will be accessible to undergraduates.