Relative interiors of finite-dimensional convex sets

Let $k=\dim (\mathrm{aff}C)$. If $k=0$, then $C$ is a singleton and is open in its affine hull. If $k>0$, the definition of affine hull supplies $k+1$ affinely independent points of $C$ spanning it. Their simplex lies in $C$ by [F1]. Its barycenter has all $k+1$ barycentric coordinates positive; sufficiently small coordinate perturbations preserve positivity and their sum one. Thus a relative neighborhood of that barycenter lies in the simplex, proving ${\mathrm{ri}}_{H}C\mathrm{\ne }\mathrm{\varnothing }$.

Fix $w,y,t$ as in the Statement. The case $t=0$ is $w\in {\mathrm{ri}}_{H}C$. Suppose $0<t<1$ and put $q=(1-t)w+ty$. Choose $r>0$ so the relative ball $B_H(w,r)$ lies in $C$. Since $y\in {\bar{C}}^H$, choose one $y_{\ast }\in C$ with $\mathrm{\parallel }{y}_{\ast }-y\mathrm{\parallel }<(1-t)r\mathrm{/}(4t)$. For every $q^{\mathrm{\prime }}\in B_H(q,(1-t)r\mathrm{/}4)$, put $w_{\ast }=(q^{\mathrm{\prime }}-t{y}_{\ast })\mathrm{/}(1-t)$. Then $\mathrm{\parallel }{w}_{\ast }-w\mathrm{\parallel }<r\mathrm{/}2$, so $w_{\ast }\in C$ and $q^{\mathrm{\prime }}=(1-t)w_{\ast }+t{y}_{\ast }\in C$ by [F1]. A relative neighborhood of $q$ lies in $C$, so $q\in {\mathrm{ri}}_{H}C$.