# FindInstance for all real values satisfying a condition

I seek at least one instance of positive fractions $$p,q,r$$ where $$p,q,r \in \mathbb{Q}^+$$, for which $$\frac12 (a - 2 b + c)^2 \ge pa^2 + qb^2 + rc^2\quad \forall\ a,b,c \in \mathbb{R}.$$ I used this code

FindInstance[{1/2 (a - 2 b + c)^2 >=
p*a^2 + q*b^2 + r*c^2 && {p, q, r} ∈
Rationals && {a, b, c} ∈ Reals }, {p, q, r}]


But it simply returns the code to me, with an error that the system contains independent variables $$a,b,c$$.

Next, I replaced FindInstance with Reduce and I got many solutions with conditions on $$a,b,c$$. I would like to restrict them to the values of $$p,q,r$$ that work $$\forall\ a,b,c$$. What is the correct way to do this?

The following shows that Reduce thinks there is no answer:

Reduce[
ForAll[
{a,b,c},
{a,b,c} ∈ Reals,
1/2 (a-2 b+c)^2>=p*a^2+q*b^2+r*c^2
] && p>0 && q>0 && r>0,
{p,q,r}
]


False

• Thank you! I will edit this further to try out my other hypotheses. – Cogicero Oct 30 '18 at 20:48
• @Carl: What does False mean in this context? – Tugrul Temel Oct 31 '18 at 14:14
• @Tugrul Based on my various "experiments" (a, b and c are vector norms), I think False means the expression cannot be reduced under the given constraints. – Cogicero Oct 31 '18 at 19:36