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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".