Let C be an arbitrary non-empty set in Rn. With a given number R and = 0 (i.e., the number is

Let C be an arbitrary non-empty set in Rn. With a given number α ∈ R and α ̸= 0 (i.e., the number α is fixed and does not change), we can define another set αC as follows:

\alpha C := {y \epsilon R^/^n | y = \alpha x, for some x \epsilon C}

 

For all the questions below, you must provide rigorous arguments to support your answer.

1. Is αC Convex?

2. Is αC a cone?

3. Now suppose that set C is convex, is αC Convex?

4. If the set C is a cone (regardless convex or not), is Is αC a cone?