Your earlier discrepancy was yet another casualty of Maple's modulus convention, where the rest have switched to parameter as the argument to EllipticE. On Tue, Oct 11, 2011 at 6:23 PM, Warren Smith <warren.wds@gmail.com> wrote:
Just by hand considering these triples (1000,1000,0) and permutations (2*pi, 4, 4) and perms (1,1,1) I observe that g1<g2+g3 is obeyed but g1<0.9999*(g2+g3) can fail;; and g1>0*(g2+g3) is obeyed but g1>0.0001*(g2+g3) can fail.
So I believe an inequality of the form g1<K*(g2+g3) is obeyed for some constant K>=1 and I do not know (but computer could pretty much tell us) what the minimum possible K is. The obvious guess is K=1.
--this crude MAPLE program
g := (a,b,c) -> (max(a,b)*EllipticE(1-min(a,b)^2/max(a,b)^2), max(a,c)*EllipticE(1-min(a,c)^2/max(a,c)^2), max(b,c)*EllipticE(1-min(b,c)^2/max(b,c)^2)); K := 0.0; do for i from 0 to 10000 do z := evalf(g(rand(), rand(), rand())): K := max(K, z[1] / (z[2]+z[3])): od; print(K); od;
output 0.9793366600 0.9992199627 0.9992199627 0.9992199627 0.9992199627
making it virtually certain that my conjecture is correct, i.e. g1 < g2+g3 the usual triangle inequality, also is valid for ellipsoid girths.
Sufficient? What are the semiaxes for girths 2,1,1? 2-epsilon, if you niggle. --rwg Addendum on Mma and that abstruse summation: Numerical checking is embarrassing. In[568]:= \[Pi]^2/12 == %564 /. b -> \[Pi] Out[568]= 2 n 2 2 2 Pi (-1) Cos[Sqrt[Pi + n Pi ]] --- == Sum[-----------------------------, {n, Infinity}] 12 2 n N[List @@ %568, 22] Out[571]= {0.822467, 0.793623 - 0.00849447 I} A violent allergy to alternating series? No, the series is positive! In[1]:= Table[Sign[((-1)^n Cos[Sqrt[\[Pi]^2 + n^2 \[Pi]^2]])/n^2], {n, 6}] Out[1]= {1, 1, 1, 1, 1, 1} Glomming terms pairwise anyway: glom[xp_] := xp /. Sum[f_, {v_, L_: 1, h_}] :> Sum[(f /. v -> 2*v - L) + (f /. v -> 2*v - L + 1), {v, L, (h + L - 1)/2}] In[4]:= glom[%%] Out[4]= 2 n 2 2 2 -1 + 2 n 2 2 2 (-1) Cos[Sqrt[Pi + 4 n Pi ]] (-1) Cos[Sqrt[Pi + (-1 + 2 n) Pi ]] Sum[--------------------------------- + ---------------------------------------------, 2 2 4 n (-1 + 2 n) 2 Pi {n, 1, Infinity}] == --- 12 In[5]:= Simplify[%, n \[Element] Integers] then crashes the kernel. Without the n \[Element] Integers (which you shouldn't need), it does nothing.