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Centrally-symmetric tight surfaces and graph embeddings. (English) Zbl 0879.53045
Let $$M$$ be a compact polyhedral (2-dimensional) surface (without boundary) in $$\mathbb{R}^d$$, which is substantial (i.e., it is not contained in a hyperplane) and tight (i.e., for every half space $$h\subset \mathbb{R}^d$$ the intersection $$M\cap h$$ is connected). It is known that examples exist for arbitrarily high $$d$$. In contrast, the author proves a sharp upper bound for the codimension in the case that the convex hull of $$M$$ is a centrally-symmetric (i.e., invariant under $$-id_{\mathbb{R}^d})$$ simplicial polytope, namely $2(d-1) (d-3) \leq 3\bigl(2- \chi(M) \bigr).$ For $$d\geq 4$$ equality holds if and only if $$M$$ is a subcomplex of the $$d$$-octahedron with $$2d$$ vertices. A second result gives a lower bound for the number of vertices of a surface triangulation admitting a fixed-point free involution.

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 52B70 Polyhedral manifolds 05C10 Planar graphs; geometric and topological aspects of graph theory
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