Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Universality theorems for inscribed polytopes and Delaunay
  triangulations   We prove that every primary basic semialgebraic set is homotopy equivalent to
the set of inscribed realizations (up to M\"obius transformation) of a
polytope. If the semialgebraic set is moreover open, then, in addition, we
prove that (up to homotopy) it is a retract of the realization space of some
inscribed neighborly (and simplicial) polytope. We also show that all algebraic
extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes.
These statements show that inscribed polytopes exhibit the Mn\"ev universality
phenomenon.
  Via stereographic projections, these theorems have a direct translation to
universality theorems for Delaunay subdivisions. In particular, our results
imply that the realizability problem for Delaunay triangulations is
polynomially equivalent to the existential theory of the reals.

30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ