  We compare the yields of two methods to obtain Bernstein type pointwise
estimates for the derivative of a multivariate polynomial in points of some
domain, where the polynomial is assumed to have sup norm at most 1. One method,
due to Sarantopoulos, relies on inscribing ellipses into the convex domain K.
The other, pluripotential theoretic approach, mainly due to Baran, works for
even more general sets, and yields estimates through the use of the
pluricomplex Green function (the Zaharjuta -Siciak extremal function). Using
the inscribed ellipse method on non-symmetric convex domains, a key role was
played by the generalized Minkowski functional a(K,x). With the aid of this
functional, our current knowledge is precise within a constant (squareroot 2)
factor. Recently L. Milev and the author derived the exact yield of this method
in the case of the simplex, and a number of numerical improvements were
obtained compared to the general estimates known. Here we compare the yields of
this real, geometric method and the results of the complex, pluripotential
theoretical approaches on the case of the simplex. In conclusion we can observe
a few remarkable facts, comment on the existing conjectures, and formulate a
number of new hypothesis.
