Suppose $X$ is a convex compact subset in $\mathbb R^n$, $n\geq 2$ and $x^*\in\mathbb R^n$ satisfying $x^*\notin X$.
For $p\in \mathbb R^n_{++}$, define the $p$-inner product by
$$\langle x,y \rangle_p= \sum_{i=1}^n x_i y_i p_i$$
and the minimum $p$-norm by
$$\pi(p,X, x^*)=\arg\min_{x\in X} \| x^*-x\|_p$$
Suppose the normal vector
$$\lambda(p)= x^*-\pi(p,X, x^*)\in \mathbb R_+^n\setminus\{0\}$$
Can I always find a sequence $(p_k)_{k\geq 1}\in \mathbb R^{n}_{++}$, and $p_k\to p$ such that
$$\lambda(p_k)= x^*-\pi(p_k,X, x^*)\in\mathbb R^n_{++}$$
for all $k\geq 1$?
Thanks.