## Decompositions of $$\mathbb R^n,n\geq 4$$, into convex sets generate codimension 1 manifold factors.(English)Zbl 1282.57028

Summary: We show that if $$G$$ is an upper semicontinuous decomposition $$\mathbb R^n,n\geq 4$$, into convex sets, then the quotient space $$\mathbb R^n/G$$ is a codimension 1 manifold factor. In particular, we show that $$\mathbb R^n/G$$ has the disjoint arc-disk property.

### MSC:

 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 57N75 General position and transversality 57P99 Generalized manifolds 53C70 Direct methods ($$G$$-spaces of Busemann, etc.)
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