On links of vertices in simplicial $d$-complexes embeddable in euclidean $2d$-space

We consider d-dimensional simplicial complexes which can be PL embedded in the 2ddimensional euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is linklessly embeddable in p2d ́ 1q-dimensional euclidean space. From this observation, we derive a new upper bound on the total number of d-simplices in an embeddable complex in 2d-space with n vertices, improving known upper bounds. Moreover, the bound is also true for the size of d-complexes linklessly embeddable in p2d` 1q-dimensional space.