All Questions

I am trying to find any techniques to analyze the measure of an image of a set under an argmax function. For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...

Fix $d>0$. Let $f:[0,\infty)\to(0,\infty)$ be a decreasing function of $x$ for $x\geq d$. Let $S_d\subset\mathbb{R}^n$ represent a set of points containing the origin such that the (Euclidean) ...