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.