Questions tagged [extreme-points]
The extreme-points tag has no usage guidance.
13 questions
1
vote
1
answer
63
views
Question on the relation between the Lagrangian Multiplier $\mathcal{L}=r+\lambda g(r,\theta_1,\dotsc,\theta_{N-1})$ and the Hessian of $r$
I want to minimize the radius $r=\sqrt{x^2_1+x^2_2+\dotsb+x^2_N}$, with the constraint $g(r,\theta_1,\dotsc,\theta_{N-1})=0$. Here $g(r,\theta_1,\dotsc,\theta_{N-1})$ is a function defined in the $N$-...
- 375
0
votes
0
answers
32
views
Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions
For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
3
votes
2
answers
218
views
Extremum placement for two-variable function
While teaching Calculus 2, one of my students asked me the following
Given 3 points $x_1$, $x_2$, $x_3$. Whether there exists one function $z = f(x,y)$ which has exactly 2 extremum and 1 saddle point:...
- 73
0
votes
1
answer
113
views
On the extreme points of two convex sets
Let $A$ and $B$ be two compact convex sets (which may be assumed to be polytopes) in $\mathbb R^n$ such that $A\cap B\ne\emptyset$. Is it then always true that either $A\cap\text{ext}B\ne\emptyset$ ...
- 128k
4
votes
0
answers
123
views
Extreme points in the space of ucp maps
Suppose $M$ and $N$ are $\mathrm{II}_1$ factors. Let $\tau\mathrm{UCP}(M,N)$ be the convex space of trace-preserving UCP maps from $M$ to $N$, equipped with the topology of pointwise weak* convergence....
- 2,840
7
votes
1
answer
312
views
Set of points covered by subspaces of small dimensions
Let $S \subset \mathbb R^d$ be finite set of points. We say that $S$ is $2$-covered if $S$ lies in a union $V_1\cup V_2$ of affine subspaces such that $\dim(V_1)+\dim (V_2)\leq d-1$. For example, if $...
- 30.6k
2
votes
1
answer
101
views
Maximizing a skew-symmetric 4D cross product
How do I find two orthonormal 4D vectors, $(x_0,x_1,x_2,x_3)$ and $(y_0,y_1,y_2,y_3)$, which maximize this function:
$-19x_1y_0 - 33x_2y_0 + 11x_3y_0 + 19x_0y_1 - 21x_2y_1 - 33x_3y_1 + 33x_0y_2 + ...
- 1,547
1
vote
0
answers
146
views
Maximizing quadratic forms
Consider the maximization problem
$$\text{maximize} \quad Q(x)= \sum_{i<j} \Big(\sum_{k} a_{ik}a_{jk}\Big) x_i x_j \quad \text{subject to} \quad \sum_{i}x_i^2=1,$$
and let $M$ be maximum value ...
2
votes
1
answer
486
views
Extreme points of an intersection of convex set with countably many linear spaces
Let $V$ be some `nice' vector space and let $T: V\to \mathbb{R}$ be a linear functional over $V$.
Define
\begin{align}
M= K \cap \bigcup_{i \in \mathbb{N} } \{ v \in V: T(v)=c_i \}
\end{align}
...
- 671
3
votes
1
answer
622
views
A Banach space where the closed unit ball is the convex hull of its extreme points
Let $X$ be a Banach space where the closed unit ball equals the convex hull of its extreme points. Is it true that this implies $X$ is reflexive?
- 633
2
votes
0
answers
230
views
Does Every Extreme Point Maximize Some Linear Functional
Let $L^2$ be the set of all square-integrable functions $f:[0,1] \to [0,1]$ and $S \subset L^2$ be a closed and convex subset of $L^2$ containing the function that is constant and equal to zero. Are ...
- 355
0
votes
1
answer
222
views
Regarding extreme point in a Banach space
Let $X$ be a Banach space. And let $X^* $ be the dual space of $X$. Let $E_X$ and $E_{X^*}$ denote the extreme points of the unit ball of $X$ and $X^*$. Let $x\in X$ and $|f(x)|=1$ for every $f\in E_{...
- 195
4
votes
2
answers
1k
views
If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am ...
- 623