Realizations of homology classes and projection areas

The relationship between convex geometry and algebraic geometry has deep historical roots, tracing back to classical works in enumerative geometry. In this paper, we continue this theme by studying two interconnected problems regarding projections of geometric objects in four-dimensional spaces: (1) Let $A$ be a convex body in $\mathbb{R}^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the areas of the six coordinate projections of $A$ in $\mathbb{R}^2$. Which tuples of six nonnegative real numbers can arise in this way? (2) Let $S$ be an irreducible surface in $(\mathbb{P}^1)^4$, and let $(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34})$ be the degrees of the six coordinate projections from $S$ to $(\mathbb{P}^1)^2$. Which tuples of six nonnegative integers can arise in this way? We show that these questions are governed by the Pl\"ucker relations for the Grassmannian $\text{Gr}(2,4)$ over the triangular hyperfield $\mathbb{T}_2$. We extend our analysis by determining the homology classes in $(\mathbb{P}^m)^n$ proportional to the fundamental classes of irreducible algebraic surfaces, resolving the algebraic Steenrod problem in this setting. Our results lead to several conjectures on realizable homology classes in smooth projective varieties and on the projection volumes of convex bodies.