Is it possible to express "either $f(x) \leq 0$ or $g(x) \leq 0$" where $f,g$ are linear constraints by using a finite number of continuous constraints/new variables, WITHOUT breaking convexity or introducing binary variables?
I know we can do it by introducing binary variables quite easily, but is it possible without them?
I think it is not. The two constraints in the OR relation can make the space non-convex (e.g. for a single variable $x$: $f(x) = x + 1, g(x) = -x - 1$, then the two parts are even disjoint so it cannot be convex).
However, you can solve the problem first using only the first constraint, then using only the other constraint and then you choose the better solution.