is this sufficient to define a simplex?

Question

I want to define a simplex based on the following properties

1. A convex polytope
2. All vertexes share an edge with all others
3. For a given vertex $v_i$ the set of all facets that the vertex belongs to is denoted by $\mathcal{F}_i$. For all vertices, the sets $\mathcal{F}_i$ are isomorphic, i.e. the sets are identical and the facets in the sets are isomorphic

For example, in the 2 dimensional regular polytopes satisfy condition 3 as each vertex belongs to two one dimensional faces that are edges and one 2 dimensional face that is the polytope

Answer 1

No. Any cyclic polytope $C(n,d)$ where $d$ is even and greater than 3 is a counterexample.

Cyclic polytopes are convex polytopes, and are $d/2$-neighborly (your condition 2 is 2-neighborliness). They are simplicial, so all the facets are isomorphic. And for even $d$, the vertex figure at each vertex is a cyclic polytope of type $C(n-1, d-1)$, so all the vertices are surrounded by the same number of facets (and all the facets are isomorphic.)