Let $A\in K_{c} (R^{p} ) $ and $u_0\in\mathbb{S} ^p$. Define $v_A:= \sup_{v\in A} \langle u_0, v\rangle$. My question is the following, there exists $\epsilon >0$ such that for all $B\in K_c(\mathbb{R} ^p) $, with $d_{H} (A, B) <\epsilon $, we have that $|v_B - v_A|<\epsilon$?.
Where $d_{H} $ is the Hausdorff distance and $\langle \cdot, \cdot\rangle$ is the dot product.
Thank you in advance.