A spherical polytope is the intersection of some closed hemispheres which is non-empty and does not contain a pair of antipodal points. A spherical complex is a tiling of the whole (d−1)-dimensional sphere by spherical polytopes. Equivalently, it is the complex obtained by intersecting a complete polyhedral fan with the a (d−1)-dimensional sphere centered at the vertex of the fan.
We know that every combinatorial information of a spherical complex can be realized as a convex polytope in dimension 3 by considering the so-called canonical polytope. The proof from Wikipedia seems to depend on the circle packing theorem and the midsphere theorem.
So my question is:
Is any spherical polytope of dimension $n$ isomorphic to a convex polytope as abstract polytope for $n > 3$?
According to This question the midsphere theorem is still unknown for higher dimension, so I suspect there is a counterexample. Any idea or reference is appreciated.
EDIT:
(1) The definition of spherical polytope is edited several times (see the discussion on comment)
(2) I just saw this wiki where it is stated that "convex polytopes in p-space are equivalent to tilings of the (p−1)-sphere." which seems to give a positive answer of my question, but no reference is provided there.
Best Answer
Another way to phrase your question is "whether every complete fan is (combinatorially equivalent to) the face fan (or the normal fan) of a convex polytopes". The answer is No in dimension $n\ge 4$ and I will provide an example derived from the following non-polytopal 4-diagram:
This picture is taken from Example 5.10 in Günter Ziegler's book "Lectures on Polytopes". The book explains why this diagram is not the Schlegel diagram of any 4-dimensional polytope. It is important to note that the "outer cell" of the diagram is a simplex.
Now, consider this diagram $D$ embedded into a 3-dimensional affine subspace of $\Bbb R^4$ that does not pass though the origin. Let $|D|$ be the simplex that forms the "outer cell" of $D$. Let $x\in\Bbb R^4$ be a point for which the convex hull $\Delta:=\mathrm{conv}(|D|\cup\{x\})$ contains the origin in its interior. This convex-hull is a 4-simplex and $|D|$ is one of its facets. Take the face fan of the simplex $\Delta$ and subdivide the cone over the facet $|D|$ into the cones over the cells of $D$. This is a complete fan in which no cone contains antipodal points. You can intersect it with a sphere to obtain a spherical complex.
Suppose now that $P$ is a polytope combinatorially equivalent to this complex. Let $y\in P$ be the vertex that corresponds to the vertex $x$ of the complex. Its four neighbors in $P$ span a simplex $\Delta'$ (that corresponds to $|D|$ in the fan above). But deleting $y$ from $P$ (i.e. taking the convex hull of all vertices of $P$ except for $y$) leaves a polytope for which $\Delta'$ is a facet (here we use that $\Delta'$ resp. $|D|$ is a simplex) and whose Schlegel diagram based at $\Delta'$ is exactly $D$. This is a contradiction.