Defeniton (Self-concordance): Let $\Omega\subseteq\mathbb{R}^n$ be a convex open set. Then $f:\Omega\to\mathbb{R}^n$ is said to be self-concordant at $x\in\Omega$ if \begin{align} (D^3f(x)[h,h,h])^2 \le 4(D^2f(x)[h,h])^3 \end{align} for all $h\in\mathbb{R}^n$. It is called $\sigma$-self-concordant barrier if $$ [Df(x)[h]]^2 \le \sigma(D^2f(x)[h,h]) $$
Note: The term $D^3f(x)(h,h,h)$ means $\sum_{i,j,k=1}^n\frac{\partial^3 f(x)}{\partial x_i\partial x_j\partial x_k}h_ih_jh_k$.
I understand that self-concordant barrier function is defined to avoid approaching to the boundary of a convex set. So the closer we are to the boundary, the larger the value of the self-concordant function is. What I do not understand is $[Df(x)[h]]^2 \le \sigma(D^2f(x)[h,h]) $, which says the derivative of the function along any direction $h$ at point $x$ has to be less than $\sigma(D^2f(x)[h,h])$. Also, I understand that $Df(x)[h]=\langle \nabla f(x) ,h \rangle$ and $D^2f(x)[h,h]=\langle \nabla^2 f(x)h ,h\rangle= h^T\nabla^2 f(x)h$ where the latter can be thought as a local norm.
First, what is the geometrical meaning of $h^T\nabla^2 f(x)h$ besides local norm? even, when it is seen as a norm and it varies from a large value to small one which characteristic of the function makes it large or small?
Second, what is the geometrical and intuition behind $ [Df(x)[h]]^2 \le \sigma(D^2f(x)[h,h]) $?
I appreciate careful thoughts which address my questions directly.