I have some troubles with the following task. We are given the equation:
$$(4t^2-9t+14)y^2 + (-20t^2x-12tz+20x+24z)y+25t^2x^2-4tz^2+11x^2+12xz+9z^2=1$$
The question is for which values of $t$ this equation represents a circle (related to this norm)? And how to compute this norm for arbitrary vector (say, $(1, 1, 1)$)?
As far as I understood, here we have to "guess" somehow what type of norm is that and then transform the LHS of the equation to the needed form, but it's not obvious how to do it. Can you give any hints about the idea of the solution? Any help would be appreciated.
Check convexity.
Indeed, the unit ball of any normed space is convex, and conversely, if a (closed/open) convex set is balanced (in case of $\mathbb{R}$ this just means symmetric about 0) and absorbing then it is the (closed/open resp.) unit ball of some seminorm.