Search Results

$\newcommand\R{\mathbb R}$For any $a\in\R$, there is no minimizer of \begin{equation*} I(f):=\int_\R xe^{f(x)-x^2}\,dx \tag{-1}\label{-1} \end{equation*} over all $f\in F_a$, where $F_a$ is the se …

Let $t:=\lambda$, so that $$R(t):=\inf_{x\in Y}F(t,x).$$ Suppose that $\int_{\partial_e Y}f\,d\mu_x$ is lower-semicontinuous in $x$ (with respect to the appropriate topology, which you appear to assum …

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\ …

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\ …

Let $X$ be a positive random variable (r.v.) with probability density function $f$. By the Cauchi--Schwarz inequality, $x_1^2=(EX)^2$ is a lower bound on $x_2=EX^2$, and this lower bound is attained i …

$\newcommand\al\alpha\newcommand\R{\mathbb R}$Let us prove the following generalization of your desired statement: Let $\psi\colon\R\to\R$ be a continuous function with compact support $S$. Let $\phi …

Yes (mainly). That $u_*:=1_{f<0}$ is a minimizer (of $\int uf$ over all $u\in X(\Omega)$) follows because, if $0\le u\le1$, then $$uf-u_*f=(u-u_*)f\ge0, \tag{1}\label{1}$$ whence $$\int u_*f\le\int uf …

Your question can be restated as follows: Suppose that for each real $a$ there is a compact set $K_a$ such that $E_{a,n}:=F_n^{-1}((-\infty,a])\subseteq K_a$ for all $n$. Does then there necessarily …

Write $t,p,t_0,P$ instead of $\theta,\psi,\theta_0,p$, respectively. In view of the condition $r<d(N,P)$, you probably meant that $t_0\le\pi/2$ (where $t_0=d(N,P)$ is the angle between the vectors $N$ …

Write $t,p,t_0,P$ instead of $\theta,\psi,\theta_0,p$, respectively. In view of the condition $r<d(N,P)$, you probably meant that $t_0\le\pi/2$ (where $t_0=d(N,P)$ is the angle between the vectors $N$ …

If $r=0$, then $f(x)=0$ for all $x\ge0$. If $r>0$, then $$f'(x)=-\frac{8 r (x+1) (8 r x+18 r+24 x+27)}{(2 x+3) (4 x+3) (2 r+2 x+3) (2 r+4 x+3)}<0$$ for $x\ge0$, $f(0)=2 \log \left(\frac{2 r}{3}+1\rig …

Let $G$ be the set of functions $g\colon\mathbb R\to\mathbb R$ such that for some strictly positive real $a$ and $b$ and all real $x$ we have $g(x)=-ax$ if $x\le0$ and $g(x)=bx$ if $x\ge0$. Let $l_1,\ …

From the context, here $AC[0, 1]$ should be the set of all absolutely continuous functions from $[0,1]$ to $\mathbb R^n$ with the norm $W^{1,1}$. Note that for all $x=(x_1,\dots,x_n)\in AC[0, 1]$ we h …

$\newcommand{\al}{\alpha}$In leo monsaingeon's answer, for the the value $J(\al)$ of the infimum it was shown that \begin{equation*} J(\al)\le9 \end{equation*} and conjectured that \begin{equation …

$\newcommand{\vpi}{\varphi}\newcommand\R{\mathbb R}$ Claim 1: The map $F$ is not Lipschitz if $p>1$. Claim 2: The map $F$ is $1$-Lipschitz if $p=1$: For all $f,g$ in $D_p$, \begin{equation*} W_ …