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Update: I have discontinued pursuing this question. However, by observing that the conjugate and its derivative are nothing more than optimum and optimizer, my question should be answered by carefully studying Chapter 7.J and the references mentioned in the commentary p.297 of [RW09]. This may be worth future research.


Let $f_1$ and $f_2$ be two uniformly convex finite functions (Chapter 3.5 of [Z02]) defined on finite dimensional space; in particular, $f_1$, $f_2:\mathbb{R}^d\to\mathbb{R}$, and for any $x,x'\in\mathbb{R}^d$ and $\alpha\in[0,1]$, there exists $q\geq 2$ such that \begin{align} f_j(\alpha x+(1-\alpha)x') \leq \alpha f_j(x)+(1-\alpha) f_j(x')-\beta\frac{\alpha(1-\alpha)}{q}\|x-x'\|_2^q.\tag{1} \end{align} for $j=1,2$. Here I use $\|\cdot\|_2$, but all norms are equivalent as the space is of finite dimension.

Defined the conjugate $f^*$ of a convex function $f$: $f^*(y):=\sup_{x\in\mathbb{R}^d}\{x^\top y-f(x)\}$. Its gradient (Proposition 11.3, p. 476 of [RW09]) is $$ \nabla f^*(y) = \mbox{arg}\min_{x\in\mathbb{R}^d}\{f(x)-x^\top y\}. $$ $\nabla f^*$ is globally Lipschitz (unsure if it is relevant here) if $f$ is uniformly convex: for any $x,x'\in\mathbb{R}^d$, \begin{align} \|\nabla f^*(x)-\nabla f^*(x')\|_2 \leq C^{1-p}\|x-x'\|_2 \end{align} for some absolute constant $C>0$ depending only on $q$ and $\beta$, where $p$ satisfies $1/q+1/p=1$. (Theorem 3.5.11(x) of [Z02] on page 218)

If $f_1$ and $f_2$ are within certain weak vicinity, e.g. for any $x\in\mathbb{R}^d$, \begin{align} |f_1(x)-f_2(x)|\leq \epsilon \|x\|_2. \tag{2} \end{align}

My Question: Do we have $\|\nabla f_1^*(y)-\nabla f_2^*(y)\|_2\leq c \epsilon\|y\|_2$ for any $y\in\mathbb{R}^d$ and some constant $c>0$ given (2)?

Related(?) results: I have found results on the convergence of a sequence of continuous uniformly convex functions $\{g_n\}_{n\in\mathbb{N}}$ by assuming the epigraphical convergence. Particularly, if $g_n\to g$ pointwise, by showing the euqi-lowersemicontinuity (p.248 of [RW09]) of $\{g_n\}_{n\in\mathbb{N}}$, $g_n\stackrel{e}{\to} g$ (Theorem 11.34 of [RW09]), where "$\stackrel{e}{\to}$" denotes the epigraphical convergence. Moreover, $g_n\stackrel{e}{\to} g$ if and only if $g_n^*\stackrel{e}{\to} g^*$, and if and only if $g_n^*\to g^*$ uniformly on every compact set (Theorem 7.17 [RW09]). However, these results are not quite suitable for my need.

Thank you for reading this post, and I would greatly appreciate if there is any comment. If my conjecture is incorrect, it would be great if my error can be pointed out.

References:

[Z02] Zalinescu, C. (2002) Convex analysis in general vector space. World Scientific.

[RW09] Rockafellar, R. T. and Wets, R. J.-B. (2009). Variational Analysis, volume 317 of Grundlehren der Mathematischen Wissenschaften. Springer, 3rd edition.

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