Let $C$ be a closed convex set in $\mathbb{R}^n$ that contains zero. Let $x \in C$ be a vector with $n$ components. Suppose we zero out one of its components. Without loss of generality, let that component be the first one, then $y=[0, x_2, \dots, x_n]^{\top}$. Is there a certain class of convex sets such that when $x \in C$ implies $y \in C$?
My try
Currently, I can think of a box that contains the zero vector, i.e., $C=[l_1,u_1]\times \dots \times [l_n, u_n]$, i.e., $l_i\leq 0 \leq u_i$ for all $i=1, \dots, n$. Also, pictorially I feel any symmetric set would do the same thing but I do not know how to show it.
Question
Are there any type of closed convex sets when $x \in C$ implies $y \in C$?