The orthant logarithmic barrier has parameter equal to its dimension

Coordinatewise inequalities show that $K$ is closed, convex, and closed under addition and nonnegative scaling. If both $z$ and $-z$ lie in $K$, every coordinate is zero, so $K$ is pointed; because $m\ge 1$, $D={\mathbb{R}}_{++}^m$ is nonempty and is the interior of $K$. If $z\in K$, then $⟨z,y⟩={\sum }_i{z}_i{y}_i\ge 0$ for every $y\in K$, so $z\in K^{\ast }$ by [F2]. Conversely, if $z\in K^{\ast }$, testing against each standard basis vector $e_i\in K$ gives $z_i=⟨z,e_i⟩\ge 0$. Hence $z\in K$ and $K^{\ast }=K$.

Direct differentiation gives $DF(x)[h]=-{\sum }_i{h}_i\mathrm{/}{x}_i$, $D^{2}F(x)[h,k]={\sum }_i{h}_i{k}_i\mathrm{/}{x}_i^2$, and $D^{3}F(x)[u,v,v]=-2{\sum }_i{u}_i{v}_i^2\mathrm{/}{x}_i^3$. The Hessian is positive definite because $D^{2}F(x)[h,h]={\sum }_i(h_i\mathrm{/}{x}_i{)}^2>0$ for $h\mathrm{\ne }0$. If $\mathrm{\parallel }h{\mathrm{\parallel }}_x<1$, then each $\mathrm{\mid }{h}_i\mathrm{\mid }\mathrm{/}{x}_i<1$, hence $x_i+h_i>0$ and the Dikin inclusion holds. If a sequence in $D$ approaches a finite boundary point, at least one coordinate tends to zero while all coordinates remain bounded above; each term $-\log x_i$ is bounded below and one tends to $+\mathrm{\infty }$, so $F\to +\mathrm{\infty }$.

Put $a_i=u_i\mathrm{/}{x}_i$ and $b_i=v_i\mathrm{/}{x}_i$. By [F1] and ${\sum }_i{b}_i^4\le ({\sum }_i{b}_i^2{)}^2$, $\mathrm{\mid }{D}^{3}F(x)[u,v,v]\mathrm{\mid }\le 2\mathrm{\parallel }a{\mathrm{\parallel }}_2({\sum }_i{b}_i^4{)}^{1\mathrm{/}2}\le 2\mathrm{\parallel }u{\mathrm{\parallel }}_x\mathrm{\parallel }v{\mathrm{\parallel }}_x^2$. Also $F(\alpha x)=F(x)-m\log \alpha$, and $\mathrm{\parallel }DF(x){\mathrm{\parallel }}_{x,\ast }^2={\sum }_i(1\mathrm{/}{x}_i{)}^2{x}_i^2=m$. These are exactly the required self-concordance and parameter conditions.