Calculus of variations

We show that the kernel is convex.

d

dx

p

1 +x^2 = p x

x^2 + 1

d^2

dx^2

p

1 +x^2 =

p

x^2 + 1 xpxx (^2) + 1
x^2 + 1 =
x^2 p+ 1 x^2
x^2 + 1
x^2 + 1 =

1

(x^2 + 1 )

p

1 +x^2

> 0.

Hence the solution is a minimum.