Self-concordant log-homogeneous barriers and their local geometry

Let $E$ be a finite-dimensional real inner-product space, let $D\subset E$ be open and convex, and let $F:D\to \mathbb{R}$ be $C^3$. Write $H_F(x)=D^{2}F(x)$ for its Hessian as a self-adjoint operator. A self-concordant barrier on $D$ is a strictly convex $F$ with positive-definite Hessian, with $F(x_k)\to +\mathrm{\infty }$ whenever $x_k$ approaches a finite boundary point of $D$, and satisfying both conditions below.

For $x\in D$, define the local norm and its dual by $\mathrm{\parallel }h{\mathrm{\parallel }}_x=\sqrt{D^{2}F(x)[h,h]},\mathrm{\parallel }\mathrm{\ell }{\mathrm{\parallel }}_{x,\ast }={\sup }_{\mathrm{\parallel }h{\mathrm{\parallel }}_x\le 1}\mathrm{\mid }\mathrm{\ell }[h]\mathrm{\mid }\mathrm{.}$ The Dikin inclusion is $\{x+h:\mathrm{\parallel }h{\mathrm{\parallel }}_x<1\}\subset D$. The self-concordance inequality is $\mathrm{\mid }{D}^{3}F(x)[u,v,v]\mathrm{\mid }\le 2\mathrm{\parallel }u{\mathrm{\parallel }}_x\mathrm{\parallel }v{\mathrm{\parallel }}_x^2(u,v\in E)\mathrm{.}$ The Newton decrement is ${\lambda }_F(x)=\mathrm{\parallel }DF(x){\mathrm{\parallel }}_{x,\ast }$.

If $K\subset E$ is a closed convex cone with nonempty interior and $D=\mathrm{int}K$, the barrier is $\nu$-log-homogeneous when $\nu >0$, $F(\alpha x)=F(x)-\nu \log \alpha$ for every $x\in D$ and $\alpha >0$, and ${\lambda }_F(x)\le \sqrt{\nu }$ for every $x\in D$. The number $\nu$ is its barrier parameter. When equality constraints $Ax=b$ are imposed, local norms and decrements are computed after restricting $D^{2}F$ and $DF$ to the tangent space $\ker A$.