##### Department of Mathematics,

University of California San Diego

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### Math 258 - Differential Geometry

## Paul Bryan

#### UCSD

## Polar Dual of Convex Bodies and Curvature Flows

##### Abstract:

Recently Ikavi has studied the evolution of a the polar dual of a convex body evolving by curvature flows. The technique has been used by Stancu and Ivaki previously in studying curvature flows are their relation to affine inequalities. I will describe how Ivaki shows the evolution of the polar dual is very similar to the evolution of the original body, evolving by an expanding Gauss curvature flow. Using techniques that go back to Tso and then later Andrews, an upper curvature bound is obtained for the original body. The novel part of this paper is that essentially the same technique yields an upper curvature bound for the dual body, which by duality corresponds to a lower curvature bound for the original body. Convergence results are then obtained in a fairly standard manner.

### October 24, 2013

### 10:00 AM

### AP&M 5829

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