probability that a random point in the real space lies inside a convex polytope

Question

I have a convex polytope obtained by the intersection of set of halfspaces. Is there any way to find the probability that a random point on real space lies inside the convex polytope. Thanks in advance.

Answer 1

There is no uniform distribution on $\mathbb R^2$; see Uniform distribution on $\mathbb Z$ or $\mathbb R$.

Answer 2

You might want to consider the following slightly changed setup: What is the probability of a random point from some encasing hypercube (of finite size!) to also fall inside your convex polytope?

Then it becomes quite easy. Simply calculate the hypervolume $V$ of your convex polytope, calculate the hypervolume $V_0$ of the hypercube, and get the searched for probability $P$ by $$P=\frac{V}{V_0}$$ --- rk