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
),