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\begin{document}$${\varvec{a}}_1,\ldots , {\varvec{a}}_r$$\end{document} be vectors in a half-space of Rn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^n$$\end{document}. We call C=a1R++⋯+arR+\documentclass[12pt]{minimal}
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\begin{document}$$C={\varvec{a}}_1{\mathbb {R}}^{+}+\cdots +{\varvec{a}}_r {\mathbb {R}}^{+}$$\end{document} a convex polyhedral cone and {a1,…,ar}\documentclass[12pt]{minimal}
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\begin{document}$$\{{\varvec{a}}_1,\ldots , {\varvec{a}}_r\}$$\end{document} a generator set of C. A generator set with the minimal cardinality is called a frame. We investigate the translation tilings of convex polyhedral cones. Let T⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$T\subset {\mathbb {R}}^n$$\end{document} be a compact set such that T is the closure of its interior, and J⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {J}}}\subset {\mathbb {R}}^n$$\end{document} be a discrete set. We say (T,J)\documentclass[12pt]{minimal}
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\begin{document}$$(T,{{\mathcal {J}}})$$\end{document} is a translation tiling of C if T+J=C\documentclass[12pt]{minimal}
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\begin{document}$$T+{{\mathcal {J}}}=C$$\end{document} and any two translations of T in T+J\documentclass[12pt]{minimal}
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\begin{document}$$T+{{\mathcal {J}}}$$\end{document} are disjoint in Lebesgue measure. We show that if the cardinality of a frame of C is larger than the dimension of C, then C does not admit any translation tiling; if the cardinality of a frame of C equals the dimension of C, then the translation tilings of C can be reduced to the translation tilings of (Z+)n\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbb {Z}}^+)^n$$\end{document}. As an application, we characterize all the self-affine tiles possessing polyhedral corners (that is, there exists a point of the tile such that a neighborhood of the point is congruent to a neighborhood of the vertex of a convex polyhedral cone), which generalizes a result of Odlyzko (Proc. Lond. Math. Soc. 37, 213–229 (1978)).
