# Convexity question

• Oct 22nd 2012, 02:23 PM
entrepreneurforum.co.uk
Convexity question
Can someone help me prove this;

I have managed to sketch them but cannot prove why its convex
• Oct 22nd 2012, 02:56 PM
FernandoRevilla
Re: Convexity question
Quote:

Originally Posted by entrepreneurforum.co.uk
but cannot prove why its convex

Consider $x=(x_i),\,y=(y_i)$ in $S_{n-1}$ and $t\in [0,1]$ then, $(1-t)x+ty=((1-t)x_i+ty_i)$ satisfies $\sum_{i=1}^n(1-t)x_i+ty_i=(1-t)\sum_{i=1}^n x_i+t\sum_{i=1}^ny_i=(1-t)+t=1$ and $(1-t)x_i+ty_i\geq 0$ for all $i=1,\ldots,n$. That is, $(1-t)x+ty\in S_{n-1}$.