} x\in B\}.$$ Hence, we have ${{\cal{M}}}B \subseteq (\rho + \frac12 (1-\rho)) B . $ The goal is to construct a convex positive definite homogeneous polynomial $p_d$ of some degree $2d$, such that its 1-sublevel set $S_d$ satisfies $$(\rho + \frac12(1-\rho)) B \subseteq int(S_d) \subset S_d \subseteq B.$$ As $S_d \subseteq B$ and ${{\cal{M}}}B \subseteq (\rho + \frac12 (1-\rho)) B$, it would follow that $${{\cal{M}}}S_d \subseteq {{\cal{M}}}B \subseteq int(S_d).$$ This would imply that $p_d(A_ix)<p(x),\forall x \in \partial S_d,$ and for $i=1,\ldots,m.$ By homogeneity of $p_d$, we get the claim in the statement of the theorem.

To construct $p_d$, we proceed in the following way. Let $$C=\{x\in \mathbb{R}^n |~|x|\leq \rho + \frac{3}{4}(1-\rho)\}.$$ To any $x\in \partial C,$ we associate a (nonzero) dual vector $v(x)$ orthogonal to a supporting hyperplane of $C$ at $x$. This means that $\forall y \in C,\ v(x)^Ty\leq v(x)^Tx$. Since $x\in {{{\rm int} }}{B},$ the set $$S(x)=\{y\in \mathbb{R}^n|~v(x)^Ty>v(x)^Tx \mbox{ and }|y|=1\}$$ is a relatively open nonempty subset of the boundary $\partial B$ of our unit ball. Moreover, $\frac{x}{|x|}\in S(x).$ Now, the family of sets $S(x)$ is an open covering of $\partial B,$ and hence we can extract a set of points $x_1,\dots,x_N$ such that the union of the 