Let F(x, y) be a binary form with integer coefficients, of degree n > 2 and nonzero discriminant D_F. Let A_F denote the area of the fundamental domain {(x, y) in R^2 : |F(x, y)| ≤ 1}. Back in the 90s my academic brother, Michael Bean, proved that the quantity |D_F|^{1/n(n-1)} * AF achieves its maximum over all forms specified above when F(x, y) = xy(x-y). Moreover, when n is fixed, he conjectured that the maximum must be attained by the form

Fn*(x,y) = prod(k=1, n, sin(kπ/n)*x – cos(kπ/n)*y)

I will talk about my recent work on this conjecture, as well as about a similar problem which concerns bounding the quantity h_F^2/n *A_F from below. Here h_F an appropriately chosen height of F.