# Tag Info

Accepted

### Relationship between KL, chi-squared, and Hellinger

This inequality is false in general. E.g., let $f$ and $g$ be pdf's on $[0,1]$ given by the formulas $f=1$ and $g=f+t\,1_{(0,1/2)}-t\,1_{(1/2.1)}$ for $t\in(0,1)$. Then the left- and right-hand sides ...
• 80.3k

The result follows from operator monotonicity of inverse over symmetric positive definite matrices, i.e., $A \succeq B \Rightarrow A^{-1} \preceq B^{-1}$, which is a special case of the Löwner-Heinz ...
• 1,293
Accepted

The proposed result holds true. I am assuming throughout that $spd$ means symmetric positive definite and that the matrices $A$ and $B$ are $n$-by-$n$ matrices over $\mathbb{R}$ for some $n > 0$. ...
• 6,453

• 80.3k
Accepted

### How do I integrate this inequality that appears in a paper of Rabinowitz?

For any unit vector $u$ and real $t>0$, let $$h(t):=H(tu).$$ Then h'(t)=H'(tu)\cdot u=\frac{H'(tu)\cdot(tu)}t\ge\frac{\mu H(tu)}t=\frac{h(t)}t. ...
• 80.3k
Accepted

### Coarea formula for measure of epsilon neighbourhood

Your formula is true (up to appropriate constants) if you take $\nu$ to be the Minkowski content, assuming that the volume of the tubular neighborhood is locally Lipschitz as a function of the tube ...
• 1,966
1 vote
Accepted

### For which value of $C(f)$ would the following inequality hold?

The question is not well posed, as it is not quite clear in what terms you want $C(f)$ to be expressed. If one only uses the terms you did mention -- the Lipschitz constant and the size of the domain, ...
• 80.3k
1 vote

### Strengthened version of Isoperimetric inequality with n-polygon

The keyword you are looking for is "quantitative isoperimetric inequalities". The case of polygons was solved in the following paper: https://arxiv.org/abs/1402.4460 The "quantitative&...
• 1,855
1 vote
Accepted

### Lower bound for KL divergence of bounded densities and $L_{2}$ metric

$\newcommand{\ep}{\varepsilon}$As in your post and comments, suppose that $f$ and $f_0$ are supported on a compact set $S$, and \begin{equation*} a\le f\le b,\quad a\le f_0\le b \end{equation*} on ...
• 80.3k

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