Is $S = \{ (x^T Q_1 x, x^T Q_2 x) \mid \|x\|_2 = 1 \}$ convex?

Question

I tried to investigate the convexity of set

$$S = \{ (x^T Q_1 x, x^T Q_2 x) \mid \|x\|_2 = 1 \}$$

where $x \in \mathbb{R}^n$, $Q_1$ and $Q_2$ are arbitrary $n \times n$ symmetric matrices, and $n > 2$. For the sake of intuition, I've plotted the set with a simple code that has the following result:

It seems that $S$ is a convex set, but I'm looking for a clear mathematical proof.

Answer 1

$S = \{(x^TQ_1x, x^TQ_2x) : \ x^Tx = 1\}$ is a convex compact set in $\mathbb{R}^2$ for $n\ge 3$. For $n=2$, the statement is not true. See Theorem 2.1 in [1], or Theorem 14.1 (page 89) in [2].

Also see: Fact 8.14.11 (page 498) in [3].

Reference

[1] L. Brickman, "On the fields of values of a matrix", Proc. Amer. Math. Soc., 12:61–66, 1961.

[2] A. Barvinok, "A Course in Convexity", Graduate Studies in Mathematics, vol. 54, 2002.

[3] D. S. Bernstein, "Matrix Mathematics: Theory, Facts, and Formulas", 2009.