Formulas for $\arg\max$

Question

are any formulas for $\arg\max(f(x))$ known
(in the context of this question, $\max(f(x))$ shall denote the essential supremum of $f(x)$ over some given domain $\Omega\subset X$)?

The reason for asking is two-fold

• due to a result of Kantorowitsch the essential supremum (or, vrai max, as he called it) can be epressed as $$\lim_{p\to\infty}\left(\int_\Omega\left|f(x)\right|^p\right)^\frac{1}{p}$$ cf e.g. this link, but I couldn't find an expression for $\arg\max(f(x))$ not even for single elements of that set.

• It seems counter intuitive, that it should not be possible to retrieve elements of $\arg\max(f(x))$ from an expression that yields the essential supremum and thus "should know" where $f(x)$ attains its "maximal" values. I would therefore be surprised to learn that that kind of intuition had not yet been investigated.

In the context of this question, the essential supremum (resp. vrai max is defined as here and it is further assumed, that $$0\ <\ f(x_0) = \lim_{\epsilon\to 0}vrai \max \left(f\left(x\in U_\epsilon(x_0)\right)\right)\ <\ +\infty\ \forall x_0\in\bar\Omega$$

I am looking for expressions for (single) elements of $\arg \max(f(x))$, e.g. the one that yields the lexical maximum of $(f(x),x)$.

Answer 1

$ \newcommand{\valArg}{\mathop{\rm arg_{val}}\nolimits} \newcommand{\essSup}{\mathop{\rm sup_{ess}}\nolimits} \newcommand{\essArg}{\mathop{\rm arg_{ess}}\nolimits} $

As there seem to be no kown formulas for calculating elements of arg max and, as the existence of such a formula would seem to come as a surprise, I'd like to present such a formula for discussion.
Provided the formula isn't demonstrated entirely wrong and interest is expressed, I can also provide ideas for proving its correctness for certain classes of functions.

Before presenting the formula, some clarification regarding terminology is necessary.
Especially different flavors of argument-sets need to be defined; to this end I introduce new operators with the following meaning:

$$ \begin{align} & \essSup(f_{|\Omega})\quad := \inf\left\{y\in\mathbb{R}\ |\ \mu\left(f_{|\Omega}^{-1}\left(y,+\infty\right)\right)=0\right\} \\ & \valArg(f_{|\Omega},y) := \left\{x\in\Omega\ |\ f(x)=y\right\} \\ & \essArg(f_{|\Omega},y) := \left\{x\in\overline{\Omega}\ |\ lim_{\epsilon\to 0}\essSup\left(f_{|\Omega\cap U_{\epsilon}(x)}\right)=y\right\} \end{align}$$
and it is the second set of arguments, that is compatible with the formula of Kantorowitsch and also applicable in cases, where the definition of $\valArg$ is too strict and renders $\valArg\left(f_{|\Omega},\essSup(f_{|\Omega})\right)$ empty.

Now, assuming
$$\begin{align} 0 & \lt f_{|\Omega} \lt +\infty \\ 0 & \lt g_{|\Omega} \lt +\infty\quad \wedge\quad x\not = y \Leftrightarrow g(x) \not = g(y) \end{align} $$

a candidate formula for an essential argument to a function's essential supremum is this one:

$$\essArg\left(\essSup(f_{|\Omega})\right)\quad\ni\quad g^{-1}\left(\lim_{\epsilon\to 0} \frac{\essSup\left(\left(f+\epsilon g\right)_{|\Omega}\right)-\essSup\left(f_{|\Omega}\right)}{\epsilon}\right) $$

The benefit of essential arguments is that one has a positive chance of finding arguments in the vincinity, whose function value comes arbitrarily close the value of the essential supremum.

The formula may only be the tip of an iceberg of further investigations, especially in view of pathological functions and their approximation via sequences of well behaved functions.
What may also be possible, is to disqualify conjecture, that are related to the location of zeros, as axioms.
Besides that, it would also be a nice application of the Gâteau derivative