##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Imre B\'ar\'any

#### London and Renyi Institute

## The minimum area convex lattice $n$-gon

##### Abstract:

Let $A(n)$ be the minimum area of convex lattice $n$-gons. (Here lattice is the usual lattice of integer points in $R^2$.) G. E. Andrews proved in 1963 that $A(n)>cn^3$ for a suitable positive $c$. We show here that $\lim A(n)/n^3$ exists. Our computations suggest that the value of the limit is very close to $0.0185067\ldots$. It turns out further that the convex lattice $n$-gon $P_n$ with area $A(n)$ has elongated shape: After a suitable lattice preserving affine transformation $P_n$ is very close to the ellipsoid whose halfaxis have length $0.00357n^2$ and $1.656n$. This is joint work with Norihide Tokushige.

Host: Van Vu

### February 17, 2005

### 3:00 PM

### AP&M 6438

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