Quantitative Version of Jensen's Inequality? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:29:26Z http://mathoverflow.net/feeds/question/119838 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119838/quantitative-version-of-jensens-inequality Quantitative Version of Jensen's Inequality? unknown (google) 2013-01-25T15:11:12Z 2013-01-25T18:53:02Z <p>Hi,</p> <p>I've been looking at situations where Jensen's inequality is almost tight, and found myself proving a lemma that I'm nearly certain exists somewhere in the literature.</p> <p>The specifics are as follows: </p> <p>Suppose we have some convex, increasing function $f(x)$ and a set of $n$ real numbers $x_i$. Define $$\delta := \frac{\sum f(x_i)}{ n } -f\left(\frac{\sum x_i}{n}\right)$$ We know that $\delta$ is positive by Jensen's, and that it is zero when all the $x_i$'s are equal to the average. Let $\delta$ be positive, but small. We now fix an epsilon, and ask how many $x_i$'s are either greater than $\frac{\sum x_i}{n}(1 + \epsilon)$ or smaller than $\frac{\sum x_i}{n}(1 - \epsilon)$. If we call that set $I$, the lemma would state that $$|I| \leq g(\delta, \epsilon) n$$ for $g$ continuous, vanishing as delta goes to zero for any fixed $\epsilon$, and depending only on the choice of $f$. What this shows is that if the Jensen's "deficit" is small, then the number of entries that are "far away" from the average is $o(n)$.</p> <p>Is this some well known (or even not well known, but existent...) lemma? </p> <p>Thanks!</p> <p>-Matan</p> <p>EDIT: Made a silly mistake in defining $\delta$ - the body should now contain the correct normalization (Thanks Daniel!)</p> http://mathoverflow.net/questions/119838/quantitative-version-of-jensens-inequality/119875#119875 Answer by Dirk for Quantitative Version of Jensen's Inequality? Dirk 2013-01-25T18:53:02Z 2013-01-25T18:53:02Z <p>This is not an answer but it may be helpful to know that there exists the notion of <em>modulus of convexity</em> of a convex function $f:X\to ]-\infty,\infty]$ defined on a Banach space $X$ which quantifies how convex a convex function is. It is defined as <code>$$\delta_f(t) = \inf\{\tfrac{1}{2}f(x) + \tfrac12 f(y) - f(\tfrac{x+y}{2})\ :\ \|x-y\|=t \}.$$</code> If $\delta_f(t)>0$ for $t>0$ then $f$ is <em>uniformly convex</em>, if $\delta_f(t)>Ct^p$ for some $C>0$, then $f$ one says that $f$ has a <em>modulus of convexity of power type $p$</em>.</p>