Let $\mathbb{R}^d$ be $d$-dimensional Euclidean space

Let $\Delta=\{x\in\mathbb{R}^d_+:\sum_{i=1}^dx^i\leq1\}$ ($x^i$ is the i-th coordinate of $x$)

(Equivalently, $\Delta$ is the convex hull of $\{(0,\cdots,0),(1,0,\cdots,0),(0,1,0,\cdots,0),\cdots,(0,\cdots,0,1)\}$)

Let $A=\{x_1,\cdots,x_k\}\subset\Delta$

Let $y\in\Delta$

Consider the set $C\equiv\{z\in\mathbb{R}^d:z=y+\sum_{i=1}^k\lambda_i(x_i-y)\mbox{ for some }(\lambda_i)_{i=1}^k\mbox{ such that }\lambda_i\geq0\ \forall i\}$

That is, $C$ is ($y$ + the hull of nonnegative linear combinations of $A-y$)

Note that $C$ is a closed convex cone with apex at $y$

I have two questions related to computational geometry.

Q1: How to compute $C$? I think $C$ can be expressed as the intersection of finitely many half-spaces (because $C$ is a polyhedron, right?). Is there an algorithm to find out those half-spaces?

Q2: How to compute the set of all extreme points of $\Delta\cap C$? I think there are only finitely many extreme points, right? Is there an algorithm?

I have no background in computational geometry. I'd love to see any reference, book or paper, so I can cite. Thanks for any help!