5.12. SADDLE POINTS∗ 147

lim infn→∞

tnH (g(y+ tn (z− y)) ,z)

≤ lim infn→∞

((1− tn)H (g(y+ tn (z− y)) ,y)+tnH (g(y+ tn (z− y)) ,z)

)≤ lim sup

n→∞

H (x,y+ tn (z− y))≤ H (x,y)

This shows that x̂ = g(y) because this holds for every x. Since tn → 0 was arbitrary, thisshows that in fact

limt→0+

g(y+ t (z− y)) = g(y) ■

Now with this preparation, here is the min-max theorem.

Definition 5.12.4 A norm is called strictly convex if whenever x ̸= y,∥∥∥∥x+ y2

∥∥∥∥< ∥x∥2 +∥y∥2

Theorem 5.12.5 Let E,F be Banach spaces with E having a strictly convex norm.Also suppose that A ⊆ E,B ⊆ F are compact and convex sets and that H : A×B→ R issuch that

x→ H (x,y) is convex

y→ H (x,y) is concave

Assume that x→ H (x,y) is lower semicontinuous and y→ H (x,y) is upper semicontinu-ous. Then

minx∈A

maxy∈B

H (x,y) = maxy∈B

minx∈A

H (x,y)

This condition is equivalent to the existence of (x0,y0) ∈ A×B such that

H (x0,y)≤ H (x0,y0)≤ H (x,y0) for all x,y (5.9)

called a saddle point.

Proof: One part of the main equality is obvious.

maxy∈B

H (x,y)≥ H (x,y)≥minx∈A

H (x,y)

and so for each x,maxy∈B

H (x,y)≥maxy∈B

minx∈A

H (x,y)

and sominx∈A

maxy∈B

H (x,y)≥maxy∈B

minx∈A

H (x,y) (5.10)

Next consider the other direction.Define Hε (x,y)≡ H (x,y)+ ε ∥x∥2 where ε > 0. Then Hε is strictly convex in the first

variable. This results from the observation that∥∥∥∥x+ y2

∥∥∥∥2

<

(∥x∥+∥y∥

2

)2

≤ 12

(∥x∥2 +∥y∥2

),

5.12. SADDLE POINTS* 147lim inf tH (8 (y+ tm (z—y)) 2)wo (1-m)H(g(y+in(z—y)),¥)< lim int ( ) )nso \ +tpH (g(yttn(z—y)),z< lim sup H (x,y +t (z—y)) < H (x,y)nooThis shows that # = g(y) because this holds for every x. Since t, — 0 was arbitrary, thisshows that in factjim g(yt+t(z— y))=s(y) aNow with this preparation, here is the min-max theorem.Definition 5.12.4 4 norm is called strictly convex if whenever x # y,“yt|< lel, IlTheorem 5.12.5 Let E ,F be Banach spaces with E having a strictly convex norm.Also suppose that A C E,B C F are compact and convex sets and that H : A x B > R issuch thatx — H (x,y) is convexy > H (x,y) is concaveAssume that x + H (x,y) is lower semicontinuous and y + H (x,y) is upper semicontinu-ous. Thenmin max H (x max min AH (xxEA yeEB ( y= yEB xcA ( y)This condition is equivalent to the existence of (x0, yo) € A X B such thatH (x0, y) < H (x0, yo) < H (x, yo) for all x,y (5.9)called a saddle point.Proof: One part of the main equality is obvious.max H (x,y) > H (x,y) > mind (x,y)yeB xEAand so for each x,max H (x,y) > maxminH (x,y)yeB yEB xEAand somin max H (x,y) > max mind (x, y) (5.10)x€A yEeB yeEB xEANext consider the other direction.Define H, (x,y) =H (x,y) +€ ||x||? where € > 0. Then H, is strictly convex in the firstvariable. This results from the observation that*< (abe) <5 (IP +IbIP),x+y2