Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means there is some $c>0$, s.t. $$\forall N>0, \enspace x_1,...,x_N \in S,\enspace v\in\mathbb{R}^N: \quad \sum_{i=1}^N \sum_{j=1}^N u(x_i,x_j)v_iv_j \geq c\left(\sum_{i=1}^N v_i\right)^2. \quad (*)$$ Of course this is satisfied for $u$ being uniformly bounded from below $u\geq c>0$. I would guess ($*$) is merely necessary for boundedness from below but only sufficient on the diagonal ($u(x,x)\geq c$ f.a. $x\in S$) and not on whole $S\times S$. Now I have two questions concerning this kind of functions:

Is there some symmetric and smooth $u$ fulfilling ($*$), that is not bounded from below?

Assuming ($*$) is it possible to show positivity on some open neighborhood $O$ of $S$, i.e. $$\forall N>0, \enspace z_1,...,z_N \in O,\enspace v\in\mathbb{R}^N: \quad \sum_{i=1}^N \sum_{j=1}^N u(z_i,z_j)v_iv_j \geq 0$$ or are there maybe counterexamples?

Thanks a lot!