I'm trying to prove that an ellipsoid defined by $$E:=\{(x,y,z) \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1 \}$$ with $a,b,c \gt 0$ is convex, but I'm having trouble with it. I tried with the definition of convexity, but I reach a point of $$ \lambda \left( \frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} + \frac{x_3^2}{c^2} \right) + 2 \lambda ( 1 - \lambda) \left( \frac{x_1y_1}{a^2} + \frac{x_2y_2}{b^2} + \frac{x_3y_3}{c^2} \right) + (1 - \lambda)^2 \left( \frac{y_1^2}{a^2} + \frac{y_2^2}{b^2} + \frac{y_3^2}{c^2} \right) \le 1 $$
and I don't know how to proceed from there (or even if it's right). I think that there might be another (easier) way to do this, but i can't come up with it, so if someone could help me with this proof I'll be really grateful $:)$
First, I think the first coefficient is $\lambda^2$.
The first and third terms you know are less than or equal to 1. The second term is also less than 1. Then all you have is:
$$\left( \frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} + \frac{x_3^2}{c^2} \right) + 2 \lambda ( 1 - \lambda) \left( \frac{x_1y_1}{a^2} + \frac{x_2y_2}{b^2} + \frac{x_3y_3}{c^2} \right) + (1 - \lambda)^2 \left( \frac{y_1^2}{a^2} + \frac{y_2^2}{b^2} + \frac{y_3^2}{c^2} \right) \le \lambda^2+2\lambda-2\lambda^+1-2\lambda+\lambda^2=1$$