# Minimum of $x^2+y^2+z^2-2xyz$

I'm looking for the minimum of this function for real numbers, I proved that the only possible local minimum is $$0$$ but I didn't find a way to prove/disprove that it's a global minimum.

• Are the variables assumed to positive? – Dr. Sonnhard Graubner Jun 26 at 14:38
• no the function is defined for all reals – wostysums Jun 26 at 14:39

There is no minimum if $$x=y=z=t\rightarrow +\infty$$ than your expression is $$3t^2-2t^3 \rightarrow -\infty$$