A question about asymptotic affinity and strict convexity with unbounded means

Question

Let $F:[0,\infty) \to [0,\infty)$ be a $C^1$ strictly convex function.

Let $\lambda_n \in [0,1],a_n\le c<b_n \in [0,\infty)$ satisfy $$ \lambda_n a_n +(1-\lambda_n)b_n=c_n \tag{1}$$ and assume that $c_n \to ֿ\infty$. (which implies $b_n \to ֿ\infty$). $c>0$ is just some constant, to make $a_n$ bounded.

Set $D_n=\lambda_nF(a_n)+(1-\lambda_n)F(b_n)-F\big(c_n\big) $, and assume that $\lim_{n \to \infty}D_n=0$

Question: Does $\lambda_n \to 0$?

My intuition is that even if $F$ becomes "less convex" (closer to being affine) when $x \to \infty$, then we cannot put to much weight on the $a_n$-since otherwise we get hit by the "convexity gap" between $a_n$ and $b_n$ by a non-negligible amount, which should make $D_n$ large.

Edit:

This is an attempt to understand Ron P's answer:

We have $D(a_n,c_n,b_n)=\lambda_n F(a_n)+(1-\lambda_n)F( b_n)-F(c_n)$, where $ \lambda_n a_n +(1-\lambda_n) b_n=c_n$.

Similarly, $D(a,c_n,b_n)=\tilde \lambda_n F(a)+(1-\tilde \lambda_n)F( b_n)-F(c_n)$, where $ \tilde\lambda_n a +(1-\tilde \lambda_n) b_n=\tilde c_n$.

Suppose that $a_n \to a$. (This implies $\lambda_n-\tilde \lambda_n \to 0$). We have

$$D(a_n,c_n,b_n)-D(a,c_n,b_n)=\lambda_n F(a_n)-\tilde \lambda_n F(a)+(\tilde \lambda_n-\lambda_n)F(b_n). \tag{2}$$ The first term tends to zero, since $F(a_n) \to F(a)$ and $\lambda_n-\tilde \lambda_n \to 0$.

Why does the second term tend to zero? we don't have control over $F(b_n)$, right?

Answer 1

The answer is yes.

Indeed, by rescaling, without loss of generality (wlog) $c=1$. To simplify the notations, let $f:=F$, $a:=a_n$, $b:=b_n$, $c:=c_n$, $t:=\lambda_n$, $D:=D_n$. Passing to a subsequence, wlog $a\to a_*\in[0,1]$ and $t\to t_*\in(0,1]$. Also, wlog $a+2\le c$, since $a\le1$ and $c\to\infty$. Also, wlog $b>c$, since wlog $t>0$ and $c>a$.

By the convexity of $f$ and inequalities $a+1\le a+2\le c$, \begin{equation*} f(a+1)\ge f(c)+\frac{a+1-c}{b-c}\,(f(b)-f(c)).\tag{1} \end{equation*} Using now the convexity of $f$ again together with the inequality $a+2\le c$ and (1), we have \begin{align*} 0\le d&:=\frac{f(a)+f(a+2)}2-f(a+1) \\ &\le \frac{f(a)}2+\frac12\,\frac{(c-a-2)f(a)+2f(c)}{c-a}-f(a+1) \\ &=\tilde d:=\frac{(c-a-1)f(a)+f(c)}{c-a}-f(a+1) \\ &\le\frac{(c-a-1)f(a)+f(c)+D}{c-a} \\ &\ \ -\Big(f(c)+\frac{a+1-c}{b-c}\,(f(b)-f(c))\Big) \\ &=\tilde D:=\frac{b-a-1}{b-a}\frac Dt\sim\frac D{t_*}\to0, \end{align*} so that \begin{equation*} d\to0. \tag{2} \end{equation*} On the other hand, \begin{equation*} d\to\frac{f(a_*)+f(a_*+2)}2-f(a_*+1)>0 \end{equation*} by the strict convexity of $f$. This contradicts (2). $\Box$

Remark: As seen from the above, condition $c\to\infty$ can be relaxed to $\liminf(c-a)>0$.

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The above solution might look somewhat mysterious. In fact, the idea is a rather simple geometric one. For any real $A,B,C$ such as $A\le B\le C$, let the "gain" $g(A,B;C)$ denote the distance between the point on the graph of the convex function $f$ with abscissa $C$ and the point with the same abscissa on the chord connecting the points on the graph of $f$ with abscissas $A$ and $B$.

So (see the picture below), $D=g(a,b;c)$ and $\tilde d=g(a,c;a+1)$, where $\tilde d$ is as defined in the above multi-line display. In that display, it was shown that $\tilde d\le\tilde D$, which is clear from the picture. Also, if $t$ is bounded away from $0$ -- that is, if $c/b$ is bounded away from $1$, then, as it is clear from the picture by looking at the similar triangles, we have $\tilde D\asymp D\to0$; cf. the last line of the above multi-line display. This and the inequality $\tilde d\le\tilde D$ imply $\tilde d\to0$.

By the convexity of $f$, for any fixed real $A,C$ such as $A\le C$, the gain $g(A,B;C)$ is nondecreasing in $B\in[C,\infty)$ (here you may want to draw another picture). Therefore and because $a+2\le c$, we have $d=g(a,a+2;a+1)\le g(a,c;a+1)=\tilde d$, so that $d\le\tilde d$, which was shown in the first three lines of the above multi-line display.

This is the geometric explanation of (1) and the above multi-line display.

Answer 2

First let's reformulate the question. For $0\leq a\leq c\leq b$, let $\lambda=\lambda(a,c,b)\in[0,1]$ be the number such that $c=\lambda a + (1-\lambda)b$, and for $f\colon \mathbb R_+\to\mathbb R$ define $$ D_f(a,c,b)= \lambda f(a)+(1-\lambda)f(b)-f(c). $$

Lemma 1. Let $f\colon \mathbb R_+\to\mathbb R$ be strictly convex and continuously differentiable. Let $0\leq a_n\leq c_n\leq b_n$ be sequences such that $a_n$ is bounded, $c_n-a_n$ is bounded away from 0, and $\limsup \lambda(a_n,c_n,b_n)>0$. Then, $\limsup D_f(a_n,c_n,b_n)>0$.

We first apply a sequence of reduction steps that allow us to assume wlog that $a_n=0$, $c_n\geq 1$, , for all $n$, and $\liminf\lambda(a_n,c_n,b_n)>0$. If you trust that that is possible, you may skip directly to Lemma 5 below.

By taking a sub-sequence $n'$ on which $\liminf \lambda(a_{n'},c_{n'},b_{n'})>0$, Lemma 1 follows from Lemma 2.

Lemma 2. Let $f\colon \mathbb R_+\to\mathbb R$ be strictly convex and continuously differentiable. Let $0\leq a_n\leq c_n\leq b_n$ be sequences such that $a_n$ is bounded, $c_n-a_n$ is bounded away from 0, and $\liminf \lambda(a_n,c_n,b_n)>0$. Then, $\limsup D_f(a_n,c_n,b_n)>0$.

By further taking a sub-sequence $n'$ on which both $a_{n'}$ converges, Lemma 2 follows from Lemma 3.

Lemma 3. Let $f\colon \mathbb R_+\to\mathbb R$ be strictly convex and continuously differentiable. Let $0\leq a_n\leq c_n\leq b_n$ be sequences such that $a_n\to a$, $c_n-a_n$ is bounded away from 0, and $\liminf \lambda(a_n,c_n,b_n)>0$. Then, $\limsup D_f(a_n,c_n,b_n)>0$.

For any fixed $\epsilon>0$, the functions $\lambda(a,c,b)$ is continuous in $a$ uniformly in $c$ and $b$ over the domain $\epsilon\leq a +\epsilon\leq c\leq b$; therefore, under the assumptions of Lemma 3, $0<\liminf\lambda(a_n,c_n,b_n)=\liminf\lambda(\lim a_n,c_n,b_n)$. Furthermore, for $\lim a_n <a<\liminf c_n$ small enough, we have $\liminf\lambda(a,c_n,b_n)>0$. Since $D_f(a,c,b)$ is decreasing in $a$, $\limsup D_f(a_n,c_n,b_n)\geq\limsup D_f(a,c_n,b_n)$. Therefore, Lemma 3 follows from Lemma 4.

Lemma 4. Let $f\colon \mathbb R_+\to\mathbb R$ be strictly convex and continuously differentiable. Let $0\leq a\leq c_n\leq b_n$ be sequences such that $c_n-a$ is bounded away from 0, and $\liminf \lambda(a,c_n,b_n)>0$. Then, $\limsup D_f(a,c_n,b_n)>0$.

Let $T\colon \mathbb R\to\mathbb R$ be the affine transformation that maps $a$ to $0$ and $\inf c_n$ to $1$. Replacing $f$ by $F=f\circ T^{-1}$, and $a,c_n,b_n$ by $T(a),T(c_n),T(b_n)$ respectively, Lemma 4 follows from Lemma 5.

Lemma 5. Let $F\colon \mathbb R_+\to\mathbb R$ be strictly convex and continuously differentiable. Let $1\leq c_n\leq b_n$ be sequences such that $\liminf \lambda(0,c_n,b_n)>0$. Then, $\limsup D_F(0,c_n,b_n)>0$.

Proof of Lemma 5. We assume wlog that $F(0)=0$ and denote $\lambda_n=\lambda(0,c_n,b_n)$ and $D_n=D_F(0,c_n,b_n)$.

Define a function $G\colon [1,\infty)\times (1,\infty)\to \mathbb R$ by $$ G(x,y)=\tfrac 1 y F(xy)-F(x). $$

Claim 6. $G$ is positive and increasing in both $x$ and $y$.

Proof of Claim 6. Since $F$ is strictly convex, $F(0)=0$, and $x = 1/y(xy)+(1-1/y)0$, $G(xy)>0$. Since $F'$ is increasing, we have $\frac {d}{dx}G(xy)=F'(xy)-F'(x)>0$, so $G$ increases in $x$. Since $F'$ is increasing and $G(x,y)=1/y\int_0^yF'(xt)x\,dt - F(x)$, $G$ increases in $y$, completing the proof of Claim 6.

Suppose there is $\lambda_0>0$ such that $\lambda_n\geq \lambda_0$ for all $n$. Then, $$ D_n = G(c_n,1/(1-\lambda_n))\geq G(1,1/(1-\lambda_0)>0, \quad\text{for all $n$.} $$ QED