On locally concave functions on simplest non-convex domains ^††thanks: Supported by the Russian Science Foundation grant 19-71-10023.

The aim of the present paper is to extend the theory of [21]. The main result of that article said that certain Bellman functions of two variables coincide with minimal locally concave functions in the case when their domain is a set theoretic difference of two unbounded convex sets, the smaller lying strictly inside the larger one. The proof relied upon a special class of $\mathbb{R}^{2}$-valued martingales and the notion of monotonic rearrangement. We improve these results in two directions: we allow our Bellman functions to depend on more than two variables and also work with the case when the domain is bounded (and therefore, not simply connected in dimension 2). While the special martingales work perfectly in this setting, the notion of monotonic rearrangement is, seemingly, not applicable. We substitute it with the notion of homogenization of a function from [20].

The work is technical: we mostly combine the ideas and methods of two cited papers. Our main results are Theorems 4.4 and 5.3. Sections 2, 3, 4, and 5 contain definitions, examples, descriptions of previous development of the theory, and statements of the results. Sections 6, 7, and 8 contain the proofs. We also place two auxiliary results in Section 9.

We wish to thank Vasily Vasyunin for his attention to our work.

Let $\Omega_{0}$ be a non-empty proper open convex subset of $\mathbb{R}^{d}$, here $d$ is a natural number. Usually $d\geqslant 2$. Let $\Omega_{1}$ be another open convex set such that $\operatorname{cl}\Omega_{1}\subset\Omega_{0}$. It will be convenient to use the notation

We assume $\Omega_{1}$ is non-empty for convenience (the case of empty $\Omega_{1}$ may be considered by means of classical convex geometry). Let $\operatorname{I}\subset\mathbb{R}$ be an interval. Consider the class of $\mathbb{R}^{d}$ valued summable functions $\varphi$ defined by the domains $\Omega_{0}$ and $\Omega_{1}$:

Here and in what follows we use the notation

for the average of a summable function $\varphi$ over a measurable set $E$ whose Lebesgue measure satisfies the requirement $0<|E|<\infty$. Sometimes we will call the points $\langle{\varphi}\rangle_{{}_{\operatorname{J}}}$, $\operatorname{J}$ being a subinterval of $\operatorname{I}$, the Bellman points of $\varphi$. Now we will show how several useful classes of functions may be described using particular choices of $\Omega_{0}$ and $\Omega_{1}$.

Let $f\colon\partial\Omega_{0}\to\mathbb{R}$ be a Borel measurable locally bounded function. We wish to find sharp estimates of the expression $\langle{f(\varphi)}\rangle_{{}_{\operatorname{I}}}$ when $\varphi\in\boldsymbol{A}$. Note that a priori it is unclear whether the integral in question exists. The function

is well defined in the case where $f$ is bounded from below (though this function may attain the value $+\infty$). The function

is a meaningful object for any $f$ locally bounded from below. Note that all the Bellman functions in the paper do not depend on the choice of $I$, because the classes of functions $\boldsymbol{A}$ (see (2.2)) and the Bellman functions themselves are defined in terms of averages. We will often omit the symbols $\Omega$ and $f$ in the notation for our Bellman functions and simply write $\boldsymbol{B}$ and $\boldsymbol{B}^{\mathrm{b}}$. We also use the notation $\Omega^{*}=\operatorname{cl}\Omega_{0}\setminus\operatorname{cl}\Omega_{1}$. Of course, we expect that the functions $\boldsymbol{B}$ and $\boldsymbol{B}^{\mathrm{b}}$ coincide in reasonable situations. However, the proof of this assertion might be unexpectedly difficult.

Now we pass to the examples and show how the Bellman functions above help to find sharp constants in various inequalities. We do not provide many details for the first two examples since they are discussed in the cited papers.

Here and in what follows we will be using the convention that concave functions may attain infinite values, see [14]. It is important that we do not allow the value $-\infty$ (see a pathological example at the end of this section). Locally concave functions play important role in the Bellman function theory; see [5] for applications to geometry. In the definition below, $\partial_{\mathrm{fixed}}\omega$ is the set of all points $x\in\omega$ such that there does not exist a segment $\ell\subset\omega$ with $x$ lying in the interior of $\ell$; the latter set is called the fixed boundary (because we fix the boundary values of locally concave functions on this set). The remaining part of the boundary is called the free boundary and is denoted by $\partial_{\mathrm{free}}\omega$. In our usual examples of a lens $\Omega=\operatorname{cl}\Omega_{0}\setminus\Omega_{1}$, we have $\partial_{\mathrm{fixed}}\Omega=\partial\Omega_{0}$ and $\partial_{\mathrm{free}}\Omega=\partial\Omega_{1}$. From now on we will be using this notation.

We are ready to define the pointwise minimal locally concave function

(note that $\mathfrak{B}_{\omega,f}(x)\in\Lambda_{{\omega},{f}}$ if this function does not attain the value $-\infty$) and state our first main theorem. The notation $\partial_{\mathrm{free}}\Omega\in C^{2}$ means that locally $\partial_{\mathrm{free}}\Omega$ coincides with a graph of a $C^{2}$-smooth function. Recall $\Omega^{*}=\operatorname{cl}\Omega_{0}\setminus\operatorname{cl}\Omega_{1}$.

This theorem generalizes, up to minor modifications, the main result of [21]; in that paper we had $d=2$ and unbounded $\Omega_{1}$ (and, therefore, unbounded $\Omega_{0}$). The methods of [21] relied upon the notion of monotonic (non-decreasing) rearrangement, which, seemingly, does not exist in the case where $\Omega$ is not simply connected. A good replacement for monotonic rearrangements was found in [20]. The disadvantage of this new method is that it does not allow to work with $x\in\partial_{\mathrm{free}}\Omega$. The main theorem in [21] states identity (4.2) for all $x\in\Omega$. Seemingly, in the larger generality Theorem 4.4 is also true for $x\in\Omega$, however, the methods from [20] do not allow to prove it (see Section 9 below for an explanation).

We need to survey the notions from [21] we will be using. We will be working with discrete time martingales adapted to a filtration $\mathcal{F}=\{\mathcal{F}_{n}\}_{n}$. We refer the reader to [15] for the general martingale theory and present here the simplified definitions we will use.

By a filtration we mean a sequence of increasing set algebras (we consider finite algebras only), that is if $A\in\mathcal{F}_{n}$, then $A\in\mathcal{F}_{n+1}$ as well. A sequence $\{M_{n}\}_{n}$ of random variables taking values in some linear space is called a martingale adapted to $\mathcal{F}$ provided, first, each $M_{n}$ is $\mathcal{F}_{n}$-measurable, and second, for each $n$ we have $M_{n}=\operatorname{\mathbb{E}}(M_{n+1}\mid\mathcal{F}_{n})$. Note that since our algebras are simple, we may freely work in infinite dimensional spaces (in the case of general martingales, there are difficulties with the definition of conditional expectation, see Section $1.3$ in [6]). All martingales we will be working with have the limit value $M_{\infty}\in L_{1}$, a random variable that generates the martingale:

Here we should take care, $M_{\infty}$ should attain values in a finite dimensional space since we wish to omit the theory of integration of functions taking values in infinite-dimensional spaces.

The main property of $M_{\infty}$ may be restated: for any atom $a\in\mathcal{F}_{n}$ (by an atom of a set algebra $\mathcal{F}$ we mean a set $a\in\mathcal{F}$ of positive measure that is minimal by inclusion), one may restore the value of $M_{n}$ on this atom by the formula

We cite an important definition from [21]. See Fig. 4 for an example.

Sometimes, we will need to use a slightly modified notion of an $(\omega,\mathfrak{D})$-martingale introduced in [20].

Consider two other Bellman functions on $\omega$: the first one is defined for $f\colon\partial_{\mathrm{fixed}}\omega\to\mathbb{R}$ measurable and bounded from below,

and the second one is defined for arbitrary locally bounded from below measurable $f$:

Recall that a martingale $M$ is called simple if there exists a natural number $N$ such that $M_{k}=M_{N}$ for all $k\geqslant N$. We will use Theorem $2.21$ from [21]. It uses the notion of a strongly martingale connected domain. A set $\omega\subset\mathbb{R}^{d}$ is called strongly martingale connected if for every $x\in\omega$ there exists a simple $\omega$-martingale $M$ that starts at $x$, that is $M_{0}=x$.

Remark $2.22$ of the same paper says that if $f$ is only locally bounded from below, $\mathfrak{B}_{\omega,f}$ is continuous at the points of the fixed boundary, then $\mathfrak{B}=\mathcal{B}^{\mathrm{b}}$. We will need a slightly stronger statement. Let us introduce yet another Bellman function

Clearly, $\mathcal{B}^{\mathrm{s}}\leqslant\mathcal{B}^{\mathrm{b}}\leqslant\mathcal{B}$. We claim that this chain of inequalities often turns into a chain of equalities.

A simple modification of the proof above allows to prove a version of Theorem 4.9.

We will also need a technical statement in the spirit of Proposition $2.7$ in [21]. The proof is also very similar, so, we omit it.

Surprisingly, the assertion of Lemma 4.10 becomes false without the assumption $\omega$ is strongly martingale connected. We use a little bit of complex variable notation to indicate points in the plane. Let $\omega$ be given by the rule

The main feature of the specific numbers in (4.11) is that the three small circles almost touch (see Fig. 5). By definition, $\partial_{\mathrm{fixed}}\omega=\{|z|=1\}$. Let $f\equiv 0$ on the fixed boundary. In this case,

On the other hand, Lemma $\mathrm{C}.5$ in [21] says $\mathfrak{B}_{\omega,f}\geqslant 0$ (the domain $\omega$ is a cheese domain in the terminology of that paper). Therefore, $\mathfrak{B}\equiv 0$ and does not coincide with $\mathcal{B}^{\mathrm{s}}$. The effect of similar nature arises when one defines rank-one or separate convex hulls, see [12].

During the proof, we will need to work with functions defined on the circle $\mathbb{T}$ of unit length (in other words, its radius equals $1/(2\pi)$). Let us equip $\mathbb{T}$ with the natural arc length measure. We will think of functions on $\mathbb{T}$ as of periodic functions on the line, i. e., identify the function $\varphi\colon\mathbb{T}\to\mathbb{R}^{d}$ with its periodic version $\varphi\hbox{\tiny{per}}\colon\mathbb{R}\to\mathbb{R}^{d}$, here

Consider a version of the class (2.2) formed from summable $\mathbb{R}^{d}$-valued functions on $\mathbb{T}$:

Note that in this definition we require the ’boundedness of oscillation’ over arcs that may wind around the circle several times. The additional domain $\hat{\Omega}_{1}$ is mostly needed for technical purposes, it seems to be unavoidable in Lemma 6.1 below.

Recall $\Omega^{*}=\operatorname{cl}\Omega_{0}\setminus\operatorname{cl}\Omega_{1}$.

Consider the Bellman functions

and

Similar to (3.1) and (3.2), we define the function (5.3) for $f$ that is bounded from below, whereas the function $\boldsymbol{B}^{\circ,\mathrm{b}}$ is a meaningful object for any $f$ locally bounded from below. Since $\varphi_{\hbox{\tiny{per}}}|_{[0,1]}\in\boldsymbol{A}$ for any $\varphi\in\boldsymbol{A}^{\circ}$, we have

The theorem below is our second main result.

The geometric functions on the right hand sides of Theorems 4.4 and 5.3 are closely related.

Before we pass to the proofs, we need to survey the theory from [20].

Note that if $\zeta_{1}$ and $\zeta_{2}$ are equidistributed, then $\operatorname{\mathbb{E}}f(\zeta_{1})=\operatorname{\mathbb{E}}f(\zeta_{2})$ for any function $f$ such that one of these mathematical expectations exists. If $\varphi\in\boldsymbol{A}^{\circ}$ and $\operatorname{J}\subset\mathbb{R}$ is an interval, then by $\mu_{\varphi|_{\operatorname{J}}}$ we denote the distribution of the random variable $\varphi_{\hbox{\tiny{per}}}|_{\operatorname{J}}$. In other words,

Note that $\mu_{\varphi|_{\operatorname{J}}}$ is a probability measure on $\partial_{\mathrm{fixed}}\Omega$. Consider the set $\mathcal{M}(\partial_{\mathrm{fixed}}\Omega)$ of all probability measures on $\partial_{\mathrm{fixed}}\Omega$ with bounded first moment. The set $\mathcal{M}(\partial_{\mathrm{fixed}}\Omega)$ is a convex subset of the space of all finite signed measures on $\partial_{\mathrm{fixed}}\Omega$ with bounded first moment. Let $\mathfrak{W}$ be a subset of $\mathcal{M}(\partial_{\mathrm{fixed}}\Omega)$. We will always impose two conditions on this set:

Note that $\mathfrak{D}\subset\partial_{\mathrm{fixed}}\mathfrak{W}$. We will be working with simple martingales that attain their values in the space $\mathcal{M}(\partial_{\mathrm{fixed}}\Omega)$; these martingales are easy to define since a simple martingale attains values in a finite dimensional linear space. We denote by $\mu_{\varphi}$ the distribution of the function $\varphi$ itself, i. e., $\mu_{\varphi}=\mu_{\varphi|_{[0,1]}}$.

The theorem above will serve as a tool to construct functions $\varphi\in\boldsymbol{A}^{\circ}$ with prescribed distributions. We must say about the difference between the formulations above and in [20]. We require weaker openness condition on $\mathfrak{W}$ here, in [20] the set $\mathfrak{W}$ was open in weak-* topology. One may go through the proof in [20] and realize that everything happens in the finite dimensional space spanned by the values of $\mathbb{M}_{\infty}$. We will apply the theorem to the sets

Note that the intersection of $\mathfrak{W}$ with any finite dimensional space $V$ spanned by delta measures is open. By definition, the condition $\varphi\in\boldsymbol{A}^{\circ}$ is almost equivalent to the condition that $\mu_{\varphi|_{J}}\in\mathfrak{W}$ for any subinterval $J\subset\mathbb{R}$ (the word ‘almost’ corresponds to the presence of the set $\hat{\Omega}_{1}$ in (5.2)). We will later show that any simple $\Omega$-martingale $M$ generates a simple martingale $\mathbb{M}$ with values in $\mathcal{M}(\partial_{\mathrm{fixed}}\Omega)$ that satisfies the conditions of Theorem 5.6 with $\mathfrak{W}$ given in (5.11). This observation will allow us to construct a function $\varphi\in\boldsymbol{A}^{\circ}$ that has the same distribution as $M_{\infty}$ (see Lemma 6.2 below).

The proof will be based on two lemmas below.

We will call the domain $\tilde{\Omega}$ as in lemma above an extension of $\Omega$.