# Basic Solutions and Extreme Points

$Let \hspace{1 mm} S = \{x:x^Tx \hspace{1 mm} \leq1\}.\hspace{1 mm}Prove\hspace{1 mm}that\hspace{1 mm}the\hspace{1 mm}extreme\hspace{1 mm}points\hspace{1 mm}of\hspace{1 mm}S\hspace{1 mm}are\hspace{1 mm}the\hspace{1 mm}points\hspace{1 mm}on\hspace{1 mm}its\hspace{1 mm}boundary.$
So $x$ is an extreme point of $S$ iff it is a basic feasible solution.
Would proof by contradiction be appropriate? Assume that the extreme points of $S$ are not on the boundary?
If I have understood correctly your question, for any linear function $f:S\to \mathbb{R}$ , $f(x)=\sum_{j=1}^nc_jx_j$ (not all $c_j\neq 0$ ) we have no critical points in $x^tx<1$ . On the other hand, $S$ is compact so, being $f$ continuous on $S$ we have absolute maximum and minimum on $S$. Necessarily this must occur on the boundary of $S$ : $x^tx=1$ .