Neighbor simplex solutions

Hello,

I am trying to locally approximate the solution space around the optimal solution x* of an LP. More exactly lets call P the polyhedron of the solution space of the LP, I am looking for a set of points X \in P such that the polyhedron P and conv(X+x*) have locally the same facet around x* : any facet of P which passes through x* is also a facet of conv(X+x*) .

One of the things that would help me a lot would be to obtain all the vertices of P adjacent to x* (in other words all points which can be accesses from x* in one simplex iteration). 

My question is : Is there a way to compute with gurobi all the “adjacents” of a solution x* (which I guess would not be too costly) or is there a simple work around? Other than doing all the calculations manually.

Many thanks,

Francois