1. Without loss of generality we can assume that hypersurface is give by
H := {(x1, ..xn) : xn = 0}
Let π : Rn → R such that π(x1, ..., xn) = xn. Then we have half space
given by
H1 := {a ∈ Rn : π(a) > 0} and H2 := {a ∈ Rn : π(a) < 0}
Now take a ∈ H1 and b ∈ H2. Line joining a and b is given by
γ(t) = a(1− t) + bt
then γ(0) = a and γ(1) = b. Take map π ◦ γ, then π ◦ γ(0) = π(a) > 0
and π ◦γ(1) = π(b) > 0. Hence by intermediate theorem, there exist some
t0 ∈ [0, 1] such that π ◦ γ(t) = 0 That is we have
π(γ(t0)) = 0.
This means line γ cuts hyperplane when t = t0 that is at γ(t0).
2. f is convex function then we will prove f ′ is non decreasing:
for x1 < x2, there is some x such that x1 < x < x2, Let λ =
x2−x
x2−x1 , then
1− λ = x−x1x2−x1 , and we have
f(λx1 + (1− λ)x2) ≤ λf(x1) + (1λ)f(x2)
We have...