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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!

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Not a direct answer, but a few references that might lead you to the appropriate literature.

The intersection of $k$ polytopes discusses "the intersection of finitely many half-spaces":

Fukuda, Komei, Thomas M. Liebling, and Christine Lütolf. "Extended convex hull." Comput. Geom. 20, no. 1-2 (2001): 13-23. (Elsevier link.)
Abstract. In this paper we address the problem of computing a minimal representation of the convex hull of the union of $kH$-polytopes in $\mathbb{R}^d$. [...]


          PolyTopInt
          (Image from Fukuda et al..)


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