Let $f\in C^\infty(\mathbb R^n;\mathbb R)$ and assume that
$f(0)=0$, $f(x)>0$ for every $x\in K\setminus \{0\}$ (K compact) and $\partial^2_{i,j}f(0)>0$ ( but $f$ need not be convex in $K$).
I want to show that there exists a quadratic function $q(x)>0$ such that $f(x)\geq q(x)$ for every $x\in K$. Taylor expansion gives
$f(x) = \sum_{i,j} \ f_{i,j}(x) \ x_i x_j$
with
$f_{i,j}(x) := \int_0^1 (1-s) \ \partial^2_{i,j}f (sx) ds$
so in the strictly convex case one can conclude immediately. What is the fastest way to proceed in the general case? I thought to split $K$ into two parts, $K_1$ containing $0$ where $f$ is strictly convex and $K_2$ (not containing $0$) where $f(x)> C>0$. This gives me quadratic functions $q_1$ and $q_2$ and I take the smallest one.
Thanks for help.