Search Results

$\newcommand\de\delta\newcommand{\KL}{\operatorname{KL}}\newcommand{\p}{\,\|\,}$The maps $$\mu\mapsto\sqrt{\KL(\mu\p\nu)}$$ and $$\nu\mapsto\sqrt{\KL(\mu\p\nu)}$$ are not convex in general. Indeed, le …

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon}$ Take any probability measures $P_0,P_1$ absolutely continuous with respect (w.r.) to $Q$. We shall prove the followi …

$\newcommand\R{\mathbb R}$Update: Within the previously established framework, we now show that it is enough to have just $N=4$ points. This improves what seems to be the previous record, of $N=5$. It …

Direct calculations show that $$(f_2*f_0)(y)=\frac{1}{4} \sqrt{\frac{\pi }{2}} e^{-\frac{y^2}{2}} \left(y^2+1\right), $$ $$(f_1*f_0)(y)=\frac{1}{2} \sqrt{\frac{\pi }{2}} e^{-\frac{y^2}{2}} y, $$ $$(f …

Let $B_+,B_1,B_2$ denote the closed unit balls w.r. to $\|\cdot\|_+,\|\cdot\|_1,\|\cdot\|_2$, respectively. Let $C$ be the convex hull of $B_1\cup B_2$. Let $\bar C$ be the closure of $C$ (w.r. to the …

$\newcommand\la\lambda$The answer is no. E.g., suppose that $X=\ell^\infty$, $z_1=e_1$, $z_2=-e_1+e_2$, and $z_k=e_2/2+e_k/k$ for $k\ge3$, where $(e_1,e_2,\dots)$ is the standard basis of $\ell^\infty …

Yes: Just take $u(x,y):=v(x)$, which will be assumed in what follows. Indeed, one can use Green's formula to show this, as is done in Christian Remling's answer. More generally, the result holds for a …

Let $n=1$, $f(t)=t^2 + |t|^{7/2}\sin(1/|t|)$ for $t\ne0$, $f(0):=0$. Then $f'(0)=0$ and $f''(0)=2>0$, so that $0$ is a strict local minimum of $f$. However, $f''(t)\sim-|t|^{-1/2}\sin(1/|t|)$ as $t\to …

$\newcommand\de\delta$If the function $f$ were convex, then a convolution of $f$ with (say) an even positive mollifier would do. However, $f$ is not convex. Yet, it would be quite easy to construct ju …

Actually, your desired conclusion does "follow just from $D$ having full measure". Indeed, without loss of generality, $U=(-1,1)^2$. Let $$X:=\{x\in(-1,1)\colon|D_x|=2\},$$ where $D_x:=\{y\in(-1,1)\co …

Indeed, this can be proved more simply, and in greater generality -- assuming only that the support of $P$ is contained in $C$ (rather than in $\mathcal X$). Indeed, without loss of generality the aff …

For any real numbers $u,v,c$ such that $u\le c\le v$, let $\mu_{c;u,v}$ denote the unique probability distribution on the set $\{u,v\}$ with mean $c$. Your generalization of Jensen's inequality follow …

Suppose the contrary, so that $g(s)<0$ for some $s\in(a,b)$. Replacing now $a$ and $b$ by $\max\{t\in[a,s)\colon g(t)\ge0\}$ and $\min\{t\in(s,b]\colon g(t)\ge0\}$, respectively, we see that without l …

We have \begin{equation} f_2(x):=f''(x)\Big/\frac{2^{x-3}}{125 \left(2^x+1\right)^2}= -x \ln2-2 \left(2^x+1-500 \ln2\right) \end{equation} and \begin{equation} f''_2(x)=-2^{1 + x} \ln^2 2<0, \en …

Let $P_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c_1,\dots, c_d$ are the same as those of the standard normal distribution $\gamma$. By Theorem 3.1 in [1 …