Geometry Seminar


Continua of minimal capacity and

Smale's mean value conjecture


Dr. Edward Crane

University of Bristol, UK & IMR Junior Fellow, HKU




Smale's mean value inequality is a constraint on the location of critical values and critical points of a complex polynomial P with degree d ³ 2.

It gives a lower bound for |P'(z)| in terms of the gradients of chords on the graph of P from the point (z, P(z)) to the stationary points (z, P(z)), where P'(z) = 0.

The mean value conjecture concerns the multiplicative constant in the inequality.  It says that the example z = 0, P(z) = zd - z should be extremal among polynomials of degree d.

We will describe a recent improvement in the constant, which relies on results of Jenkins and Kuzmina giving a lower bound for the logarithmic capacity of a plane continuum containing three given points.



June 4, 2008 (Wednesday)


2:30 – 3:30pm


Room 517, Meng Wah Complex, HKU




All are welcome