How to proof that the set of all $X$ such that $X.A{\ge} c$ to some real number c is convex?

Question

How can i proof the following statement:

" Let $\mathrm A\in \mathbb R^{n}$ and $\mathrm c\in \mathbb R$, the set $\mathbb S$ of all elements belonging to $\mathbb R^{n}$ and satisfying the condition: $\mathrm X.A\ge c$ is a convex set. "

I tried using the property of convex sets that this set should contains all members between two another distincts members, but i could not. Does anyone know a way??

Thank you. ${}{}{}{}{}$

Answer 1

Let $X,Y\in S$ and $0<\lambda<1$. Then, $$((1-\lambda)X+\lambda Y)\cdot A=(1-\lambda)(X\cdot A)+\lambda(Y\cdot A)\ge (1-\lambda)c+\lambda c=c$$ Thus, $(1-\lambda)X+\lambda Y\in S$, for all $0<\lambda<1$. So, $S$ is convex.