Question

9. (a) Show that the set {( , 3|) 5 15}

2 2

S = x y x + y ≤ is convex.

Answer 1

Let C be the convex set,

$C = { x\in R^2 | u.v\le9}$

The set C is convex if for any $u,w \in C$ the point

$tu+(1-t)w \in C$ for all $t\in[0,1].$

Let $u,w \in C$ and $t \in [0,1].$

$(tu+(1-t)w).v$

$= (tu).v+[(1-t).w].v$

$= tu.v+(1-t)w.v$

$\le t\times 9 + (1-t)\times 9$

$= 9$

Therefore, $(tu+(1-t)w).v \le 9$

Hence, C is convex.