Is $\text{ball}_{R}(X^{\ast})\cap\bigcap_{i\in I}\{x^{\ast}\in X^{\ast} | \ |x^{\ast}(x_i)|\leq r_i\}$ norm (or weak*) connected?

Question

Let $X$ be a real normed space and let $\text{ball}_{R}(X^{\ast})$ denote the closed $R$-ball ($R>0$) in the dual space $X^{\ast}$ of $X$ (w.r.t. the usual operator norm on $X^{\ast}$). Given a set $I$, a subset $\{x_i\}_{i\in I}\subset X$, and a subset $\{r_{i}\}_{i\in I}\subset[0,\infty)$. Is the set $$\text{ball}_{R}(X^{\ast})\cap\bigcap_{i\in I}\{x^{\ast}\in X^{\ast} | \ |x^{\ast}(x_i)|\leq r_i\}$$ norm (or weak*) connected? I'm working on a proof in functional analysis where I need this set to be connected w.r.t. the norm or weak* topology. Any suggestions are greatly appreciated!