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$\def\u{{\bf u}}\def\p{{\bf p}}\def\q{{\bf q}}$ Consider all the points of intersection of the lines $L_d$ with the hyperplanes $H_k$ defining the facets $F_k$. Let $\q$ be the one closest to $\p$; suppose $\q=L_d\cap H_k$. Then $(d,k)$ is a desired pair.

Firstly, $\q$ should belong to $F_k$, otherwise the segment $[\p,\q]$ would intersect the boundary of a polytope at a point on another facet; thus $d\in G_k$. Next, let $\q_1,\dots,\q_D$ be the intersection points of the hyperplane $H_k$ with the lines $L_1,\dots,L_D$ (some of these points may be ideal). Then $\|\p-\q\|=\min_i\|\p-\q_i\|$ which is equivalent to your relation.

It seems that

EDIT: Surely, the condition of convexity is unnecessarycondition IS necessary.

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$\def\u{{\bf u}}\def\p{{\bf p}}\def\q{{\bf q}}$ Consider all the points of intersection of the lines $L_d$ with the hyperplanes $H_k$ defining the facets $F_k$. Let $\q$ be the one closest to $\p$; suppose $\q=L_d\cap H_k$. Then $(d,k)$ is a desired pair.

Firstly, $\q$ should belong to $F_k$, otherwise the segment $[\p,\q]$ would intersect the boundary of a polytope at a point on another facet; thus $d\in G_k$. Next, let $\q_1,\dots,\q_D$ be the intersection points of the hyperplane $H_k$ with the lines $L_1,\dots,L_D$ (some of these points may be ideal). Then $\|\p-\q\|=\min_i\|\p-\q_i\|$ which is equivalent to your relation.

It seems that the condition of convexity is unnecessary.