I have a convex polytope defined by $\mathbf{Ax \leq b}$ (row-wise)
The log-barrier function is defined as:
$$\phi(x) =-\sum{\log(b_i - a_ix)}$$
The Hessian of the log-barrier is :
$$\nabla^2\phi(x) = \text{...magic algebra} = A^TD^2A$$
where D = diag($\frac{1}{b_i-a_ix}$)
I understand that the log-barrier is sort of a measurement of how extreme the point $x$ is from the boundary of the polytope (as $\phi$ tends to $\infty$ as $x$ tends to the boundary). However, my question is:
How do I intuitively understand this Hessian?
More background: I am trying to understand intuitively what the Dikin Ellipsoid is (https://www.cs.cmu.edu/~ggordon/10725-F12/scribes/10725_Lecture23.pdf, http://stanford.edu/class/ee364b/lectures/dikin_slides.pdf )