Unconditional constants and polynomial inequalities

If PP is a polynomial on RmRm of degree at most nn, given by P(x)=∑α∈Nm,|α|≤naαxα, and Pn(Rm)Pn(Rm) is the space of such polynomials, then we define the polynomial |P||P| by |P|(x)=∑α∈Nm,|α|≤n|aα|xα. Now if B⊆Rm is a convex set, we define the norm ‖P‖B≔sup{|P(x)|:x∈B} on Pn(Rm)Pn(Rm), and then we investigate the inequality ⦀P⦀B≤CB‖P‖B, providing sharp estimates on CB for some specific spaces of polynomials. These CB’s happen to be the unconditional constants of the canonical bases of the considered spaces.