Convex Set Examples.

Question

Is a closed interval $[a,b]$, or $[0,1]$ in particular, a convex set ? I mean, let $\lambda\in [0,1]$. Then, \begin{align} a\lambda+(1-\lambda)b&=a\lambda+b -b\lambda &=(a-b)\lambda+b \end{align}
Where does it lie?

Answer 1

if $a \le x \le b$ and $a \le y \le b$,

since $\lambda$ and $1-\lambda$ are nonnegative,

$\lambda a \le \lambda x \le \lambda b$ and $(1-\lambda )a \le (1-\lambda)y \le (1-\lambda)b$,

Summing the equations, we have

$$a \le \lambda x + (1-\lambda )y \le b$$