Given $\mathbb{S}^n_+:=\{x\in \mathbb{R}^{n+1}: |x|=1,x_{n+1}>0\}$ be the open domain in $\mathbb{S}^n$, or be viewed as the geodesic ball centered at the pole with radius $\frac{\pi}{2}$ in $\mathbb{S}^n$, where $\mathbb{S}^n$ equiped with the standard spherical metric.
My question is: **Whether there exists a smooth positive strictly convex function $f$ defining in $\mathbb{S}^n_+$?**
Here strictly convex means the Hessian of function$f:\mathbb{S}^{n}_+
\to \mathbb{R}^+$ is positive definite with respect to the spherical metric $g_{\mathbb{S}^n}$, i.e. there exists $c_0>0$ such that $\nabla^2f\geq c_0 g_{\mathbb{S}^n}$.
Another further questions： Is there exists a strictly convex function defining on the geodesic sphere (within injectivity radius) located in ambient Riemannian manifold space with nonnegative curvature?

Many thanks in advance for any comments or advice.