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I am looking for a reference for the following claim:

Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.

Suppose that "$\phi$ is convex at $c$". i.e. for any $x_1,x_2>0, \alpha \in [0,1]$ satisfying $\alpha x_1 + (1- \alpha)x_2 =c$, we have $$ \phi(c)=\phi\left(\alpha x_1 + (1- \alpha)x_2 \right) \leq \alpha \phi(x_1) + (1-\alpha)\phi(x_2) . $$

Then $\phi$ satisfies Jensen ineqaulity "at $c$".

Finite form:

Given $\lambda_i \in [0,1],x_i\in(0,\infty),i=1,\dots,k$ such that $\sum_{i=1}^k \lambda_i=1,\sum_{i=1}^k \lambda_ix_i=c$, we have $$\phi(\sum_{i=1}^k \lambda_ix_i) \le \sum_{i=1}^k \lambda_i \phi(x_i).$$

A more general probabilistic (measure-theoretic) form:

Given a random variable $X \in (a,b)$ with expectation $E(X)=c$, we have $$ \phi(c)=\phi(E(X)) \le E(\phi(X)). $$

In addition, if $\phi$ is strictly convex at $c$, then equality holds if and only if $X$ is constant a.e..

Both of these forms of Jensen inequality follow from the existence of a supporting line to the graph of $\phi$ at $c$.


The proof of the latter fact is not hard, but I couldn't find a source in the literature that presents this "localized" form of Jensen inequality, under the sole assumption of "convexity at a point". (In fact, I couldn't even find the term "convex at a point" anywhere...).

I find it impossible to believe that this doesn't show up in existing literature. Any help would be welcomed.

Comment:

Convexity at $c$ does not imply that the one-sided derivatives exist, so the standard proof for the existence of a supporting line (subgradient) does not apply here. (When the function is convex on an interval, every number between the two-sided derivatives form a subgradient).

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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 follows immediately from the well-known fact that any probability distribution $\mu$ on $\mathbb R$ with a given mean $c\in\mathbb R$ is a mixture of probability distributions of the form $\mu_{c;u,v}$. See e.g. formula (2.13).

Details: Indeed, that formula implies $$Ef(X)=\int_{S_c} Ef(X_{u,v})\,\nu_X(du\times dv)$$ for some probability measure $\nu_X$ (depending on the distribution of $X$) on the set $S_c:=\{(u,v)\in\mathbb R^2\colon u\le c\le v\}$ and all functions $f\colon\mathbb R\to\mathbb R$ such the function $\mathbb R\ni x\mapsto f(x)-kx$ is bounded from below for some some real $k$.

Now, if $f$ is convex at $c$, then $Ef(X_{u,v})\ge f(EX_{u,v})=f(c)$ for all $(u,v)\in S_c$, and hence $Ef(X)\ge f(c)$.

Answer to your additional question concerning the strict convexity at $c$: Moreover, if $f$ is strictly convex at $c$, then $Ef(X_{u,v})> f(EX_{u,v})=f(c)$ for all $(u,v)$ in the set, say $S_c^\circ$, of all $(u,v)\in S_c$ such that $u<c<v$. Hence, $Ef(X)>f(c)$ unless $\nu(S_c^\circ)=0$. On the other hand, the condition $(u,v)\in S_c\setminus S_c^\circ$ implies that $Eg(X_{u,v})=g(c)$ for all functions $g\colon\mathbb R\to\mathbb R$. So, the condition $Ef(X)=f(c)$ implies $\nu_X(S_c^\circ)=0$, which in turn implies that $Eg(X)=\int_{S_c\setminus S_c^\circ} g(c)\,\nu_X(du\times dv)=g(c)$ for all (say) nonnegative $g\colon\mathbb R\to\mathbb R$, which means that $P(X=c)=1$.

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  • $\begingroup$ Thank you for this answer. Can you please elaborate on how proposition 3.18 implies the required form of Jensen inequality? I am having a bit of trouble deducing it. BTW, this approach seems to be a bit of an overkill, isn't it? (it seems less elementary than just proving the existence of a supporting line. I hoped there would be a more elementary reference, say some book on convex analysis, rather than a paper from 2009. I am quite sure this version of Jensen was known long before that...). $\endgroup$ – Asaf Shachar Jul 9 at 5:24
  • $\begingroup$ Anyway, I am interested in understanding better this more general approach, so if you could elaborate more on how it implies Jensen, it would be great. $\endgroup$ – Asaf Shachar Jul 9 at 5:24
  • $\begingroup$ Oh, and last question: Does the approach you have described imply that if $\phi$ is strictly convex at $c$, and there is an equality in Jensen, then the random variable is constant a.e.? $\endgroup$ – Asaf Shachar Jul 9 at 8:39
  • $\begingroup$ @AsafShachar : I have added the details you requested, and even an (affirmative answer) to your additional question concerning the strict convexity at $c$. $\endgroup$ – Iosif Pinelis Jul 9 at 14:27
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    $\begingroup$ @WillieWong : This is so. Indeed, consider first the probability measures with a compact support. For any compact $K\subset\mathbb R^n$, let $P_{K,c}$ be the set of all probability measures with support $\subseteq K$ and mean $c$. Then $P_{K,c}$ is convex and compact and hence, by the Choquet–Bishop–de Leeuw theorem, any point in $P_{K,c}$ is a mixture of extreme points of $P_{K,c}$, which latter are probability measures with support of cardinality $\le n+1$. It remains to note that any probability measure on $\mathbb R^n$ is a mixture of probability measures with a compact support. $\endgroup$ – Iosif Pinelis Jul 9 at 21:12

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