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By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. Associating to each polytope the set of normal vectors $N(P) = \{\alpha_i u_i\}$ this describes a bijection between polytopes up to translation and balanced vector configurations. (This generalizes to a bijection between convex bodies up to translation and their surface measures.)

When $n=2$ and $P,Q\subset\mathbb R^2$ are polytopes, the set $N(P+Q)$ associated to the Minkowski sum is the union $N(P)\cup N(Q)$ with vectors in the same direction added together. (The sum of the surface measures.)

For $n>2$ these are two different operations: the union of the vector configurations with vectors in the same direction summed up defines the Blaschke sum $P \# Q$.

Is any description of $N(P+Q)$ in terms of $N(P)$ and $N(Q)$ known for $n>2$? Since the normal fan of $P+Q$ is the refinement of the two fans, the normal directions are indeed obtained as a union, but the facet volumes are not summed.

By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. Associating to each polytope the set of normal vectors $N(P) = \{\alpha_i u_i\}$ this describes a bijection between polytopes up to translation and balanced vector configurations. (This generalizes to a bijection between convex bodies up to translation and their surface measures.)

When $n=2$ and $P,Q\subset\mathbb R^2$ are polytopes, the set $N(P+Q)$ associated to the Minkowski sum is the union $N(P)\cup N(Q)$ with vectors in the same direction added together. (The sum of the surface measures.)

For $n>2$ these are two different operations: the union of the vector configurations with vectors in the same direction summed up defines the Blaschke sum $P \# Q$.

Is any description of $N(P+Q)$ in terms of $N(P)$ and $N(Q)$ known for $n>2$?

By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. Associating to each polytope the set of normal vectors $N(P) = \{\alpha_i u_i\}$ this describes a bijection between polytopes up to translation and balanced vector configurations. (This generalizes to a bijection between convex bodies up to translation and their surface measures.)

When $n=2$ and $P,Q\subset\mathbb R^2$ are polytopes, the set $N(P+Q)$ associated to the Minkowski sum is the union $N(P)\cup N(Q)$ with vectors in the same direction added together. (The sum of the surface measures.)

For $n>2$ these are two different operations: the union of the vector configurations with vectors in the same direction summed up defines the Blaschke sum $P \# Q$.

Is any description of $N(P+Q)$ in terms of $N(P)$ and $N(Q)$ known for $n>2$?

By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. Associating to each polytope the set of normal vectors $N(P) = \{\alpha_i u_i\}$ this describes a bijection between polytopes up to translation and balanced vector configurations. (This generalizes to a bijection between convex bodies up to translation and their surface measures.)

When $n=2$ and $P,Q\subset\mathbb R^2$ are polytopes, the set $N(P+Q)$ associated to the Minkowski sum is the union $N(P)\cup N(Q)$ with vectors in the same direction added together. (The sum of the surface measures.)

For $n>2$ these are two different operations: the union of the vector configurations with vectors in the same direction summed up defines the Blaschke sum $P \# Q$.

Is any description of $N(P+Q)$ in terms of $N(P)$ and $N(Q)$ known for $n>2$? Since the normal fan of $P+Q$ is the refinement of the two fans, the normal directions are indeed obtained as a union, but the facet volumes are not summed.