本日のC.A.D.

t が実数，a, b, c がある 3 角形の 3 辺の長さ，s=(a+b+c)/2 ならば

$a^t(s-a)+b^t(s-b)+c^t(s-c)\le(1/2)abc(a^{t-2}+b^{t-2}+c^{t-2})$

1.10 Bottema, et. al. "GEOMETRIC INEQUALITIES"

今回は指数に変数が含まれるので（比の値の範囲ではなく）大小関係の特定に十分な不等式を導入し，（過剰ですが）その辺々の比の値の範囲を求めます．具体的には，右辺を展開した形から，a^{t-1}，b^{t-1}，c^{t-1} をそれぞれ x，y，z と一般化して

$xa(s-a)+yb(s-b)+zc(s-c)\le(1/2)(bcx+cay+abz)$

また，t の符号が任意なので，x, y, z の不等式制約は正値広義単調性までとします．

? tst12([ex,ex,ex,ex,ex,ex],[d,a,b,c,x,y,z],andx,"0<a,a<=b,b<=c,c<a+b,0<x,0<z,(y-x)*(y-z)<=0,0<d,x*a*((a+b+c)/2-a)+y*b*((a+b+c)/2-b)+z*c*((a+b+c)/2-c)==d*(b*c*x+a*c*y+a*b*z)");Ans()
 *** using Lazard's method (MPP17).
[z,3]
[y,4]
[x,1]
[c,8]
[b,23]
[a,1]
[d,90]
time = 532 ms.
197 197(0,0) 5576(56,300) 32574(440,6114) 32574(0,0) 268888(0,19328) 132(0,9988) 
 *** combined adjacent 131 cells.
1[[d,1] < d <= [2*d-1,1],true,true,true,true,true,true]
time = 1min, 9,981 ms.

? tst12([ex,ex,ex,ex],[d,b,c,y,z],andx,"1<=b,b<=c,c<1+b,0<z,(y-1)*(y-z)<=0,0<d,((1+b+c)/2-1)+y*b*((1+b+c)/2-b)+z*c*((1+b+c)/2-c)==d*(b*c+c*y+b*z)");Ans()
 *** using Lazard's method (MPP17).
[z,3]
[y,4]
[c,8]
[b,23]
[d,90]
time = 423 ms.
197 5576(56,300) 32574(440,6114) 268888(0,19328) 132(0,9988) 
 *** combined adjacent 131 cells.
1[[d,1] < d <= [2*d-1,1],true,true,true,true]
time = 47,092 ms.

ここで導入した不等式は，変形すると

$0\le x(a-b)(a-c)+y(b-c)(b-a)+z(c-a)(c-b)$

つまり，Schur の不等式（https://en.wikipedia.org/wiki/Schur%27s_inequality）の仲間なので，3 角不等式の制約を外しても十分です．

? tst12([ex,ex,ex,ex,ex,ex],[d,a,b,c,x,y,z],andx,"0<a,a<=b,b<=c,0<x,0<z,(y-x)*(y-z)<=0,0<d,x*a*((a+b+c)/2-a)+y*b*((a+b+c)/2-b)+z*c*((a+b+c)/2-c)==d*(b*c*x+a*c*y+a*b*z)");Ans()
 *** using Lazard's method (MPP17).
[z,3]
[y,4]
[x,1]
[c,7]
[b,19]
[a,1]
[d,62]
time = 344 ms.
101 101(0,0) 2350(52,147) 17646(240,2436) 17646(0,0) 144248(0,9922) 60(0,6924) 
 *** combined adjacent 59 cells.
1[[d,1] < d <= [2*d-1,1],true,true,true,true,true,true]
time = 35,123 ms.

? tst12([ex,ex,ex,ex],[d,b,c,y,z],andx,"1<=b,b<=c,0<z,(y-1)*(y-z)<=0,0<d,((1+b+c)/2-1)+y*b*((1+b+c)/2-b)+z*c*((1+b+c)/2-c)==d*(b*c+c*y+b*z)");Ans()
 *** using Lazard's method (MPP17).
[z,3]
[y,4]
[c,7]
[b,19]
[d,62]
time = 357 ms.
101 2350(52,147) 17646(240,2436) 144248(0,9922) 60(0,6924) 
 *** combined adjacent 59 cells.
1[[d,1] < d <= [2*d-1,1],true,true,true,true]
time = 22,029 ms.