Show that the ellipsoid is convex

Question

Show that an ellipsoid $$\{x\in \mathbb{R}^n \ : \ x^TAx+2b^Tx+c\le 0\},$$ where $A\in \mathbb{S}^n_+$, is a convex set.

Answer 1

For $x,y$ on the ellipsoid let $z=y-x.$

For $t\in \Bbb R$ let $F(t)=(x+tz)^T A (x+tz)+2b^T (x+tz)+c.$

We have $$F(t)=t^2(z^T A z) +t(z^T Ax + x^T A z+2b^Tz)+ F(0).$$ With $x,y,z $ fixed, we have $F''(t)=2z^T A z\ge 0.$