Unconditional constants and polynomial inequalities Original Research Article

If PP is a polynomial on RmRm of degree at most nn, given by View the MathML sourceP(x)=∑α∈Nm,|α|≤naαxα, and Pn(Rm)Pn(Rm) is the space of such polynomials, then we define the polynomial |P||P| by View the MathML source|P|(x)=∑α∈Nm,|α|≤n|aα|xα. Now if View the MathML sourceB⊆Rm is a convex set, we define the norm View the MathML source‖P‖B≔sup{|P(x)|:x∈B} on Pn(Rm)Pn(Rm), and then we investigate the inequality