10
votes
Accepted
If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
Every separable Banach space $X$ can be equivalently renormed so that every point in the unit sphere is an extreme point: Take an injective bounded linear operator $T$ from $X$ into $\ell_2$ and use $|...
- 31.5k
8
votes
If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
No. Let $X$ and $Y$ be Banach spaces, and set $Z=X\oplus Y$, with $\||(x,y)|\|:=\|x\|+\|y\|$. Assume that $x$ is a extreme point of $X$ with $\|x\|=1$. Then $(x,0)$ becomes an extreme point of $Z$; ...
- 2,118
4
votes
Accepted
A Banach space where the closed unit ball is the convex hull of its extreme points
The answer is "No" because there exist nonreflexive Banach spaces in which every point on the surface of the unit ball is extreme. See, for example, Diestel, Geometry of Banach spaces, Chapter 4, ...
- 902
3
votes
Accepted
Extremum placement for two-variable function
$\newcommand\R{\mathbb R}$
$$f(x,y)=\frac{2}{x^2+(y-1)^2+1}-\frac{1}{x^2+(y+1)^2+1} \tag{1}\label{1}$$
will do.
Indeed, this function $f$ has exactly three critical points: a saddle point $(0,\approx-...
- 128k
3
votes
Accepted
On the extreme points of two convex sets
No: consider the line segments $\{0\}\times[-1,1]$ and $[-1,1]\times\{0\}$ in $\mathbb{R}^2$.
- 8,820
3
votes
Set of points covered by subspaces of small dimensions
Yes. One can take $C=2d+2$.
I'll do the projective case, so my points will be nonzero points in a $d+1$-dimensional space $W$ and I'm looking for pairs of subspaces of total dimension $\leq d+1$ that ...
- 149k
2
votes
Extreme points of an intersection of convex set with countably many linear spaces
The last paragraph is wrong. Consider the case $m=1$. $\tilde{M}$ is the intersection of $K$ with one hyperplane. In general this will not contain any extreme point of $K$, so its extreme points ...
- 54.2k
2
votes
Accepted
Question on the relation between the Lagrangian Multiplier $\mathcal{L}=r+\lambda g(r,\theta_1,\dotsc,\theta_{N-1})$ and the Hessian of $r$
If the constraint function $g$ allows to parametrize $r$ by $r = \varphi(\theta_1,\dots,\theta_{M})$ via the implicit function theorem, then you can rewrite your optimization problem as an ...
- 2,670
2
votes
Extremum placement for two-variable function
Just draw a contour plot starting with a saddle:
You can assign pretty much arbitrary value to each contour within the limits of common sense, so for an appropriate choice of values this diagram will ...
- 61.9k
2
votes
Accepted
Regarding extreme point in a Banach space
Suppose that $x$ satisfies the condition and is not extreme in the unit ball of $X$. It means that we can write $x = \frac{y+z}{2}$ for some $y,z \in B_{X}$ different from $x$. Since $|\frac{1}{2}f(y)+...
- 5,005
1
vote
Maximizing a skew-symmetric 4D cross product
Too long to comment. See if this helps.
Since the matrix (defining the quadratic) is skew-symmetric, we can first decompose it into $Q\Sigma Q^\top$, where $Q$ is orthogonal, and $\Sigma$ is a block ...
- 1,216
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