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Dimension-raising homomorphisms between lattices of convex bodies. (English) Zbl 1338.52004

Let \(K^n\) denote the class of compact convex sets in \(\mathbb{R}^n\). Equipped with the two operations \(\cap\) (intersection) and \(\vee\) defined through \[ L \vee M := \text{conv} (L \cup M), \] where conv\(( \dots )\) denotes convex hull, \(K^n\) becomes a lattice (i.e., a partially ordered set with unique meets and joins). The main theorem in the paper under review classifies the homomorphisms \(K^n \to K^{ n+1 }\): for \(n \geq 3\), a function \(\Phi : K^n \to K^{ n+1 }\) is a nontrivial lattice homomorphism if and only if there exist a hyperplane \(H \subset K^{ n+1 }\) and an affine bijection \(\phi : K^n \to H\) such that precisely one of the following cases hold:
(i)
For \(v = 0 \in \mathbb{R}^{ n+1 }\) or some fixed \(v \in \mathbb{R}^{ n+1 }\) not parallel to \(H\), \[ \Phi(C) = \bigcup_{ x \in C } [\phi(x), \phi(x)+v] \qquad \text{for all } C \in K^n . \]
(ii)
For some fixed \(o \in \mathbb{R}^{ n+1 } \setminus H\) and some fixed \(\gamma \in [0, 1)\), \[ \Phi(C) = \bigcup_{ x \in C } [\phi(x), \gamma \phi(x) + (1-\gamma)o] \qquad \text{for all } C \in K^n . \]
Homomorphisms \(K^n \to K^n\) were classified in [P. M. Gruber, Abh. Math. Semin. Univ. Hamb. 61, 121–130 (1991; Zbl 0754.52006)]; the general case \(K^n \to K^m\) remains open, though the paper under review gives a general dimension bound for \(n < m\).

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)

Citations:

Zbl 0754.52006
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References:

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