Strictly convex or “non-singular” continuous homogeneous polynomials in real or complex Banach spaces

Abstract

Fix an even integer d≥2. Let X be a separable real Banach space. Here we prove the existence a closed subspace Y of X and homogeneous degree d polynomials Q, Q 1, respectively on X and on Y with the following properties: i. Q(x) ≥ 0 for all x∈X and Q(x)=0 if and only if x∈Y; ii. Q 1(y)>0 for all y∈Y; iii. Q 1 is strictly convex, i.e. for every ε > 0 the set B(Q 1,ε):= {y∈Y: Q 1(y)≤ε} is convex and Q 1(tu+(1-t)v))<ε for all u, v∈B(Q 1,ε) and all 0 <t <1; iv. the continuous homogeneous form Q˜ induced by Q on X / Y is strictly convex. In the complex case we prove a similer existence result for complete intersections of the projects spaces P(Y) and P(X/Y).