In what sense are "projecting" and "taking sections" of polytopes dual operations? It seems to be folklore that projecting and taking sections of polytopes are somehow "dual operations" (e.g. explicitly noted in the abstract of this paper, or suggested by this answer to an MO question of mine). But I do not know the exact formal meaning of this statement.
A polytope $P\subset\Bbb R^n$ is the convex hull of finitely many points. Let $V\subset\Bbb R^n$ be an affine subspace. Denote by $\pi_V$ the orthogonal projection onto $V$.

  
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*The section of $P$ along $V$ is the polytope $P\cap V$.
  
*The projection of $P$ onto $V$ is the polytope $\pi_V P$.
  

Can someone explain to me, in what sense these operations are dual to each other? I am familiar with dual polytopes, and I assume it has something to do with these. I am not certain whether the "dual operations" make use of the same affine subspace $V$.
 A: The main idea here is that of lattice inspections. You could consider the different Delone cells of the lattice and its Voronoi cell. By definition according cell complexes are vice versas duals.
But if you would embed your lattice in a higher dimensional regular one, you would observe that the various lower dimensional Delone cells just occur as different layerwise sections of higher dimensional lattice polytopes (cells). And likewise you get that the lower dimensional Voronoi cell is obtained as the projection of that very lattice polytope.
For instance the Delone cells of (triangular) lattice $A_2$ are triangles, half of those pointing up, half of those pointing down. The Voronoi cell is the regular hexagon. - The triangular tiling and the hexagonal one clearly are vice versas duals. - But when embeding $A_2$ into (cubical) lattice $BC_3$, you see that those triangles occur as vertex layer sections of those cubes, while the hexagon is the projection of the cube with respect to the orthogonal direction of the chosen vertex layers.
--- rk
