  For a convex domain $K$ in the complex plane, the well-known general
Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$
must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in
general, $||p'||$ can be arbitrarily small as compared to $||p||$.
  The situation changes when we assume that the polynomials in question have
all their zeroes in the convex body $K$. This was first investigated by
Tur\'an, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk
$D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although
partial results provided general lower estimates of lower order, as well as
certain classes of domains with lower bounds of order $n$, it was not clear
what order of magnitude the general convex domains may admit here.
  Here we show that for all compact and convex domains $K$ with nonempty
interior and polynomials $p$ with all their zeroes in $K$ $||p'|| > c(K) n
||p||$ holds true, while $||p'|| < C(K) n ||p||$ occurs for any $K$. Actually,
we determine $c(K)$ and $C(K)$ within a factor of absolute numerical constant.
