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Feb
5
comment Minimum of the Schatten 1-norm
No. You only have $B_\lambda \to A$, but (as long as $\lambda \ne 1/2$) $B_\lambda \ne A$.
Feb
5
revised Minimum of the Schatten 1-norm
added 24 characters in body
Feb
4
comment When a set is convex, how does the polar set of its polar set equal the original?
I don't think so. The proposition itself can also be considered as a separation theorem :)
Feb
4
answered When a set is convex, how does the polar set of its polar set equal the original?
Feb
4
answered Minimum of the Schatten 1-norm
Feb
4
comment When a set is convex, how does the polar set of its polar set equal the original?
What do you mean by "guidance"? Do you want a hint for a proof or a motivation why the statement is related with convexity?
Feb
4
comment When a set is convex, how does the polar set of its polar set equal the original?
This is not true. Additionally, the closedness of $X$ is required.
Feb
3
comment Analytical expression of the solution to parabolic equation
This equation does not have a unique solution. A solution is given by $f(x,t) = \mathrm{e}^{\lambda \, t}$.
Feb
3
comment What non-convex functions be written as the $\min$ of multiple convex functions?
@Sobi: The function $i_{\{x\}}$ is convex (its $0$ on $x$, and $+\infty$ elsewhere). Hence, the function $x' \mapsto i_{\{x\}}(x') + f(x)$ equals $f(x)$ on $x$ and elsewhere it is $+\infty$. Hence, $f(x') = \min_x \{i_{\{x\}}(x') + f(x)\}$.
Feb
3
comment What non-convex functions be written as the $\min$ of multiple convex functions?
@Rahul: I use characteristic function and indicator function the other way round as wikipedia. And I feel that 'indicator function' is much more used in the branch of convex analysis.
Feb
3
comment Separation of a weak-star closed wedge by a weak-star continuous linear functional
Which properties do you have for $A$? If $A$ is compact w.r.t. the weak-star topology, you can just invoke the above theorem.
Feb
3
comment $f$ convex strictly decreasing function , is $f'(x+\delta)-f'(x)$ convex
The convexity of $g$ would amount to $g''(x) = f'''(x+\delta) - f'''(x) \ge 0$. I do not see any reason why this should hold true.
Feb
2
comment Can one prove the existence of a fixed point for a shrinking map on a sequentially compact metric space WITHOUT proving the space is compact?
Did you tried to define $x_n = T^n x_0$ for some $x_0 \in X$ and use sequential compactness on the sequence $\{x_n\}$? I don't know if this works.
Feb
2
comment What non-convex functions be written as the $\min$ of multiple convex functions?
Every function $f$ can be written as $\min_x \{ i_{\{x\}} + f(x)\}$, where $i_{\{x\}}$ is the indicator function of $x$, which is convex.
Feb
2
comment $S(x,r)$is not convex in a normed space
In case $X = \{0\}$ your set $S$ is empty, hence convex.
Feb
2
answered If $\phi(v_1),…\phi(v_\rho)$ are linearly independent, show that $v_1,…,v_\rho$ are linearly independent
Feb
1
comment Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?
But the dirac measure is not a function.
Jan
31
comment Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?
This is not true: Find a sequence $f_n$ which converges towards a Dirac delta measure. Then, this sequence will not have a limit point in $S$. However, it converges towards the Dirac also in the Wasserstein metric.
Jan
31
comment Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?
You might want to add $f \ge 0$ in the definition of $S$.
Jan
30
comment Is closed convex set with finite number of extreme points convex polyhedron
@leducquang: Some standard references concerning (convex) polytopes are Grünbaum's "Convex polytopes" and Ziegler's "Lectures on Polytopes".