Timeline for Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$
Current License: CC BY-SA 3.0
7 events
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| Aug 13, 2016 at 16:07 | comment | added | Steve | I think that quantity is indeed dimension free because we can explicitly write the normalization constant in $\rho(y)$ as a integral. Then $int | D^{\nu} \rho(y) | dy$ can be written as a integral with respect to the $|y|$, using spherical coordinates. The quantity that involving dimension is canceled by the normalization constant and we are left with a integral on $[0,1]$, which is finite and dimension-free. | |
| Aug 13, 2016 at 15:26 | comment | added | Steve | Thanks again for your help! Another question is whether $\int |D^{\nu} \rho(y) | d y $ depends on dimension $n$. Can it be dimension free? In paper ``On the dependence of the Berry–Esseen bound on dimension'' (mii.lt/files/Bentkus_2003_2.pdf) an approximation function is constructed as $ f_{\delta} (x) = g( d(x,A)/\delta)$, where $d(x,A)$ is the distance of $x$ to set $A$ and $g$ is smooth and supported on $[0,1]$. They show that the derivative of this function is bounded by $2/ \epsilon$ and the hessian bounded by $8/\epsilon^2$. Can we prove similar things using mollifier? | |
| Aug 8, 2016 at 16:56 | history | edited | VictorZurkowski | CC BY-SA 3.0 |
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| Aug 8, 2016 at 15:43 | comment | added | VictorZurkowski | I thought you were going to ask that! My first attempt had a $\rho$ taylor to the norm, and the argument $f = 1$ in $A$ and $0$ outside $A^{\epsilon}$ could be done using balls in the p-norm. But I thought that the bounds on derivatives of $f$ would involve $n.$ May be you or someone can check, and see if there is a way out. I'll add the construction of the $\rho$ as answer. | |
| Aug 8, 2016 at 6:49 | comment | added | Steve | Thanks very much, Victor! I think the second argument about the values $I_{\mathcal{O} }* \rho_{\delta}(x)$ should be for $x \notin A + B_2(0, 2\delta)$. In addition, for general $\ell_p$ balls, you utilize the fact that the $\ell_p$ norms are equivalent in the topological sense. But the constant $C_p$ may depends on the dimension $n$. Is it possible to have something that is dimension free? | |
| Aug 7, 2016 at 21:39 | vote | accept | Steve | ||
| Aug 7, 2016 at 15:13 | history | answered | VictorZurkowski | CC BY-SA 3.0 |