  In this paper we revisit the anisotropic isoperimetric and the
Brunn-Minkowski inequalities for convex sets. The best known constant
$C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due
to Segal [\ref{bib:Seg.}]. We improve that constant to $Cn^6$ for convex sets
and to $Cn^5$ for centrally symmetric convex sets. We also conjecture, that the
best constant in both inequalities must be of the form $Cn^2,$ i.e., quadratic
in $n.$ The tools are the Brenier's mapping from the theory of mass
transportation combined with new sharp geometric-arithmetic mean and some
algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.
