Timeline for Bounding function of norms in constrained vector space

Current License: CC BY-SA 3.0

11 events

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Oct 18, 2015 at 1:33	comment	added	user		Thanks for your thoughts on the problem. I've awarded the bounty, and I'll wait a bit on accepting the answer just in case anyone has an idea for how to prove the lower bound (i.e., why vectors of the form $(1, a, b, ... b)$ achieve the minimum).
Oct 18, 2015 at 1:32	history	bounty ended	user
Oct 13, 2015 at 22:11	comment	added	user44143		L = Limit[ f /. {w -> 1 + a + n b, x -> 1 + a^2 + n b^2, y -> 1 + a^3 + n b^3, z -> 1 + a^4 + n b^4}, n -> Infinity]; S = Simplify[L, Assumptions -> 0 < a < b < 1]; NMinimize[{b, S[[1, 1, 2]], 0 < a < b < 1}, {a, b}]
Oct 13, 2015 at 21:12	comment	added	user		Ah, very good reasoning. I had also observed similar numerical bounds (and the same phenomenon for the worst-case $(1, a, b, \ldots b)$ vector for the lower bound), but hadn't been able to formalize it at all. By the way, did you get the $a = 0.176$ from calculus or from experiments?
Oct 13, 2015 at 16:44	comment	added	user44143		$f$ is of the form $(a+\sqrt{b})\,/\,c$ with $c \ge a \ge 0$. So $f \le 1$ is equivalent to $(c-a)^2 - b \ge 0$. And as it happens $(c-a)^2 -b = 4c( (wy-x^2)+(xz-y^2)-(wz-xy))$.
Oct 13, 2015 at 16:01	comment	added	user		Nice! For the upper bound, can you explain how you get $w y -x^2 \geq 0$ implies $f \leq (wy−x^2)+(xz−y^2)−(wz−xy)$? I can't quite see it...
Oct 13, 2015 at 14:08	history	edited	user44143	CC BY-SA 3.0	deleted 125 characters in body
Oct 13, 2015 at 6:20	history	edited	user44143	CC BY-SA 3.0	added 1 character in body
Oct 13, 2015 at 5:30	history	edited	user44143	CC BY-SA 3.0	added 65 characters in body
Oct 13, 2015 at 5:08	history	edited	user44143	CC BY-SA 3.0	edited body
Oct 13, 2015 at 4:51	history	answered	user44143	CC BY-SA 3.0