Showing convexity

[patter2809]

Set S defined as { (x,y)^t | y =/> x² }. Required to show that S is convex.

Let u = (x₁,y₁) with u E S, and v = (x₂,y₂) with v E S. Need to show that any w, with w = a u + (1-a) v (0</= a </= 1) E S.

w = (w₁,w₂) with,

w₂ =/> w₁

(a y₁ + (1-a) y₂) =/> (a x₁ + (1-a) x₂)²

a y₁ + (1-a)y₂ =/>a² x₁² + 2 (1-a) a x₁x₂ + (1-a)² x₂²

Since 0 </= a </= 1, a² </= a.

y₁ =/> x₁² since u E S which implies ay₁ =/> a²x₁², and since v E S, (1-a)y₂ =/> (1-a)² x₂².

But this still leaves the 2 (1-a) a x₁x₂ ?

Thanks in advance!