The shape of a feasible region with equality and inequality constraints

Question

I was wondering if anyone can help me with this (probably basic) question. I want to know how the following feasible region looks like if we have thousands of variables. The constraints are linear. The region is convex and closed. Does the region have a scientific or mathematical name and specific properties (especially for finding projections of vectors on this region)?

\begin{array}{ll} & Ax = b \\ & Bx \le d \\ &x \ge 0. \end{array}

Thanks a lot for your time and help.

Answer 1

The mathematical name for this region is convex hull.

Now, how does this region look like ? It is not representable if you have thousands of variables, as we cannot see in $4D$ and above. Each variable is a dimension, so you can only draw the region if there are less than $3$ variables.

In $3D$ it would look like this.