Search Results

$\newcommand{\na}{\nabla}\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}$Let $Y$ denote the interior of $X$. Then indeed $g:=\na f$ is a homeomorphism of $Y$ onto $g(Y) …

Such a parametrization will not be in general real analytic, because the angle of the support line may be varying too slowly at some points. E.g., let the boundary of the convex set $D$ be $$C:=\{(x, …

$\newcommand\R{\mathbb R}$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 …

$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$We have $$\mu(\u+t\v+K)=f(t):=\int_{\R^n}d\x\,F(\x,t),$$ where $$F(\x,t):=\ga(\x)\ …

By the Tsirel’son--Ibragimov--Sudakov argument, reviewed on the first page in Bobkov, pushing the measure forward from the cube to the canonical Gaussian on $\mathbb R^n$ and using the Gaussian isope …

$\newcommand{\1}{\mathbf 1}\newcommand{\ep}{\varepsilon}\newcommand{\tr}{\operatorname{tr}}$The min-max value is $\sqrt n$. Indeed, take any real $n\times n$ matrix $H$ with $|\det H|=1$. By the singu …

A counterexample is given by the following conditions: $n=185$, $$a_k=\sum_{j=k}^n b_j,\quad b_j:=\frac{c_j}{j+1}, \quad c_j:=\frac{1000}7\,1(j=47)+\frac{302}7\,1(j=185).$$ Indeed, then $a_0\ge\cdots …

For finite-dimensional $V$, your definition of a face is equivalent to the definition of a poonem, according to part (i) of Exercise 7 on page 21 of the book Convex Polytopes by B. Gruenbaum; then the …

Now the answer is almost complete: modulo some extra work on the strictness of relevant inequalities, we do have the uniqueness. The additional ideas used to come to this conclusion are these: (i) to …

A counterexample is $c_1= 2,c_2= 2,c_3= 81,x_1= -8,x_2= -1,x_3= \frac{2}{9},y_1= 0,y_2= 0,y_3= 0$, $z_1= 0,z_2= 0,z_3= 0$. Added in response to the OP's comment: With the additional condition that t …

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

The answer is no. Indeed, your question can be restated as follows: Is it true that for some constant $c>0$ we have $$\int P\ln\frac PQ\,d\mu\ge c\int(\ln P-\ln Q)^2\,d\mu, \tag{1} $$ where $P$ and $ …

$\newcommand{\De}{\Delta}\newcommand{\R}{\mathbb R}\newcommand{\ga}{\gamma}\newcommand{\Ga}{\Gamma}$This is to provide a detalization on Will Sawin's comment. Specifically, let us show that the best u …

It is not true in general that $d_\Omega$ is convex, however smooth the boundary is. E.g., if $n\ge2$ and $\Omega$ is the unit ball centered at the origin, then $d_\Omega(x)=1-|x|$ is concave in $x\in …

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 …