Intereting Posts

Calculate $f^{(25)}(0)$ for $f(x)=x^2 \sin(x)$
Orthogonal Projection onto the Unit Simplex
Height and minimal number of generators of an ideal
$n$ points are picked uniformly and independently on the unit circle. What is the probability the convex hull does not contain the origin?
Inverse Laplace transform of one complicated function
Ravi substitution in inequalities
Learning Model Theory
Determining the rank of a matrix based on its minors
Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$
Prove that $H$ is a abelian subgroup of odd order
$x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$
Uniqueness of adjoint functors up to isomorphism
If we accept a false statement, can we prove anything?
How do I prove this sum is not an integer
Uniqueness of compact topology for a group

I recently came across the following problem:

Let $ K \subseteq \mathbb{R}^2 $ be a closed convex cone (meaning K is closed under non negative linear combinations) and I am asked to show it is polyhedral meaning it is the intersection of finitely many half spaces, or alternatively that it is finitely generated $ K = cone(A) $ where $ A $ is a finite set and cone(A) denotes all possible non-negative linear combinations of elements of A, or alternatively that K has a finite number of extreme vectors meaning vectors $x\in K$ such that if $ x=y+z $ where $ y,z\in K $ are non-negative multiples of x.

I have given three possible definitions of polyhedral cone but for some reason I cannot seem to see why a convex cone which is closed in $\mathbb{R}^2$ satisfies either one of the three possible definitions. I certainly need and appreciate all the help I can get on this.

- Prove a certain property of linear functionals, using the Hahn-Banach-Separation theorems
- Running average of a convex function is convex
- About the slack variable for hinge-loss SVM
- An inequality regarding the derivative of two concave functions
- Proof of Clarkson's Inequality
- Finding the dual cone

- When does it make sense to define a generator of a set system?
- Is the convex hull of closed set in $R^{n}$ is closed?
- “Support function of a set” and supremum question.
- interior points and convexity
- Relation between mean width and diameter
- Finding the dual cone
- Is every convex function on an open interval continuous?
- Infimum over area of certain convex polygons
- Is it sufficient for convexity?
- Understanding the subdifferential sum rule

Let $K$ be a closed cone in the plane. If $K$ is not contained in a closed halfspace, then $K$ coincides with the plane. Suppose then that it is contained in a closed halfspace, whose boundary is a line $L$. Let now $L’$ be a line parallel to $L$ contained in the interior of that halfspace. The intersection $K\cap L’$ is a convex closed subset of $L’$, so it is a closed interval. Now consider cases: the interval may be the whole of $L’$, a closed halfline, or a bounded closed interval.

- Prove $R \times R$ is NOT an integral domain
- Assume that $ 1a_1+2a_2+\cdots+na_n=1$, where the $a_j$ are real numbers.
- Symmetric Group $S_n$ is complete
- Generating functions for combinatorics
- Fundamental group and path-connected
- Closed form for a pair of continued fractions
- Taylor Series for $\log(x)$
- Rudin against Pugh for Textbook for First Course in Real Analysis
- Overview of basic results about images and preimages
- Relating the normal bundle and trivial bundles of $S^n$ to the tautological and trivial line bundles of $\mathbb{R}P^n$
- $n^2 + 3n +5$ is not divisible by $121$
- Product of totally disconnected space is totally disconnected?
- How does $(a,b)$ where $a,b\in \mathbb{Q}$ form a basis for $\mathbb{R}$?
- Given $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$ show that $x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$
- Prove that this function is bounded