compact and convex set

Question

I recently have worked on compact convex sets in the context of time series and my question is related to that. If we have a set $$ C=\{\beta_1 X_1 +\beta_2 X_2 +\phi H_1 , |\beta_1|+|\beta_2|+|\phi|\leq K\} $$ where $X_1$ and $X_2$ are independent and $H_1$ consists of dependent columns(and correlated to $X_1$ and $X_2$). Also $X$ and $H_1$ are bounded. Then, under which conditions can we say that C is a compact and convex set?