  The Bohnenblust--Hille (polynomial and multilinear) inequalities were proved
in 1931 in order to solve Bohr's absolute convergence problem on Dirichlet
series. Since then these inequalities have found applications in various fields
of analysis and analytic number theory. The control of the constants involved
is crucial for applications, as it became evident in a recent outstanding paper
of Defant, Frerick, Ortega-Cerd\'{a}, Ouna\"{\i}es and Seip published in 2011.
The present work is devoted to obtain lower estimates for the constants
appearing in the Bohnenblust--Hille polynomial inequality and some of its
variants. The technique that we introduce for this task is a combination of the
Krein--Milman Theorem with a description of the geometry of the unit ball of
polynomial spaces on $\ell^2_\infty$.
