Hlawka’s functional inequality

AbstractThe paper is devoted to the functional inequality (called by us Hlawka’s functional inequality)
$$f(x+y)+f(y+z)+f(x+z)\leq f(x+y+z)+f(x)+f(y)+f(z)$$for the unknown mapping f defined on an Abelian group, on a linear space or on the real line. The study of the foregoing inequality is motivated by Hlawka’s inequality:
$$\|x+y\|+\|y+z\|+\|x+z\|\leq\|x+y+z\|+\|x\|+\|y\|+\|z\|,$$which in particular holds true for all x, y, z from a real or complex inner product space.


Introduction
Let X be a real or complex inner product space and let x, y, z ∈ X be arbitrary. One can verify the classical identity x + y 2 + y + z 2 + x + z 2 = x + y + z 2 + x 2 + y 2 + z 2 . (1) A related inequality which is known as Hlawka's inequality, appeared in 1942 in a paper of Hornich [17].
In 1963 Djoković [9] proved an important generalization of (2), which is often called Hlawka-Djoković inequality; see also Adamović [2,3]. This inequality was obtained independently by Smiley and Smiley [37]. Some additional comments can be found in a paper of Simon and Volkmann [36].
In 1974 Witsenhausen [46] showed the importance of inequality (2) for the geometric properties of normed linear spaces. For a survey of other related results which were known before 1993 the reader is referred to the monograph Mitrinović et al. [26].
More recently, Wada [44] obtained a matrix version of the Hlawka-Djoković inequality. Further extensions are due to Cho et al. [8], Rǎdulescu and Rǎdulescu [32] and Honda et al. [15], among others. Integral generalizations of (2) are due to Takahashi, et al. [38], [41], among others. Janous [18] obtained several applications of the Hlawka-Djoković inequality for orthogonal polynomials. Niederreiter and Sloan [29] provided some interesting applications of Hlawka's inequality and in a recent paper of Wu [48] a version of Hlawka's inequality for fuzzy real numbers is given.
Let us also note that inequalities similar to inequality (2) are important in the theory of Aleksandrov spaces. In 2008 Berg and Nikolaev [6,Theorem 6] presented an elegant characterization of CAT (0)-spaces using a related inequality. Let us note that an alternative proof of this result was obtained by Sato [35].
A normed linear space for which inequality (2) holds for all x, y, z is called a Hlawka space (see e.g. Takahasi et al. [39,40]) or quadrilateral space (see Smiley and Smiley [37], Watson [45]). It is easy to provide an example of a Banach space which is not a Hlawka space. It suffices to consider the space R 3 with the supremum norm and to take x = (1, 1, −1), y = (1, −1, 1) and z = (−1, 1, 1). Then Modifying this example we can obtain even more: if x, y, z are the same as before and the space R 3 is equipped with the norm then, after some computations, one can verify that Hlawka's inequality (2) does not hold for every x, y, z ∈ X for any p > log 1.5 3 ≈ 2.71 (see Witsenhausen [46]).
Each inner product space is a Hlawka space (see e.g. Mitrinović et al. [26,Chapter XVIII,Section 4]). Moreover, every two dimensional space is a Hlawka space (see Kelly et al. [22]). It is also easy to observe that further examples of Hlawka spaces are L 1 or, more generally, L 1 (X, μ), where (X, μ) Vol. 87 (2014) Hlawka's functional inequality 73 is an arbitrary space with measure. Consequently, by a theorem of Lindenstrauss [24] each two-dimensional real normed linear space E is isomorphically isometric to a subset of L 1 ([0, 1]) and therefore E is a Hlawka space (this was independently proved in an elementary way in [22]). Moreover, Witsenhausen [46,Corollary 1.2] showed that the space L p (0, 1) is a Hlawka space for 1 ≤ p ≤ 2. Therefore, one can see that all Banach spaces having the property that all its finite dimensional subspaces can be embedded linearly and isometrically in the space L p ([0, 1]), with some 1 ≤ p ≤ 2 are Hlawka spaces (see Niculescu and Persson [28] and Lindenstrauss and Pe lczyński [25]). Further, Witsenhausen [47] proved that a finite-dimensional real space with piecewise linear norm is embeddable in L 1 if and only if it is a Hlawka space. However, Neyman [27] showed that in the general case embeddability in L 1 does not characterize Hlawka spaces. Concluding, to the best of the author's knowledge, no characterization of Hlawka spaces is presently known. It is worth noting that identity (1) does not hold in every Hlawka space. In fact, one can check that (1) implies the parallelogram law: (1)), which characterizes inner product spaces among all normed linear spaces (see Fréchet [10] and Jordan and von Neumann [19]).
The present paper is devoted to the functional inequality: for a real-valued unknown mapping f defined on an Abelian group or on a vector space (Sect. 2) and then, in Sect. 3, on the real line. Let us point out that this functional inequality already appeared in the year 1978 in paper [47] of Witsenhausen. A related functional equation: and also a few more general equations were studied by Kannappan [20] in 1995.
Witsenhausen [47,Lemma 1] proved that each positively homogeneous solution of (3) defined on R n is a support function of a centrally symmetric convex body (i.e. of a nonvoid compact convex set). Moreover, solutions of inequality (3) play a significant role in the characterization of zonotopes (see e.g. Witsenhausen [46,47]).
Let us note that several functional inequalities related to inequality (3) have already been discussed by other authors, mainly in connection with subadditivity and convexity. In 1965 Popoviciu [31] provided a characterization of convex mappings defined on an interval I as continuous solutions of the inequality: where n ≥ 3 and 2 ≤ k < n are fixed integers and x 1 , . . . , x n ∈ I are arbitrary. A particular case of Popoviciu's inequality (with n = 3 and k = 2) is the inequality: which plays a significant role in the theory of convex functions (see Niculescu and Persson [28]). Further results for a system of related inequalities are due to Vasić and Adamović [42], Kečkić [21], Pečarić [30], among others.
Burkill [7] considered the expression: These studies were developed further by Baston [5], a related result for twicedifferentiable real functions was proved by Vasić and Stanković [43].
Our purpose is to contribute to the above-mentioned studies. We will describe all solutions of functional inequality (3) under some additional conditions.
Note that the particular solutions of inequality (3) are f = · on a Hlawka space and f = · 2 on a Hilbert space. More generally, it is clear that for an arbitrary additive functional a : X → R and for each additive operator L : X → Y having its values in a Hlawka space or in a Hilbert space, respectively, both mappings . In Sect. 2 we prove the converse statements under some homogeneity assumptions.

Inequality (3) on linear spaces
We will deal with functional inequality (3) under some additional homogeneity assumptions. In particular we will provide conditions which are necessary and sufficient for a representation of solutions of inequality (3) as a sum of an additive functional and a norm or a square of a norm of a continuous linear operator.
A subadditive mapping f : X → R defined on an arbitrary Abelian group X which satisfies the homogeneity condition: is called sublinear. Let us recall a result of Ger [14] which characterizes sublinear mappings on Abelian groups.
Theorem 1 (Ger [14]). Assume that (X, +) is an Abelian group and f : X → R is an even and sublinear mapping. Then there exist a Banach space E and an additive mapping A : X → E such that f can be represented in the form: We begin with a lemma.

Lemma 1.
Assume that (X, +) is an Abelian group and f : X → R is arbitrary. If f satisfies functional inequality (3) jointly with f (0) = 0, then there exist an additive function a : X → R and an even function g : is satisfied for all s, t ∈ X.
Proof. Let us define a : X → R as for all x, y ∈ X. Therefore, since a is odd, a is additive. Note also that the function g = f −a solves inequality (3) and is even. Next, fix arbitrary s, t ∈ X and apply (3) for the mapping g with the substitution (x, y, z) → (s − t, 2t, s − t). We arrive at

Corollary 1.
Assume that (X, +) is an Abelian group and f : X → R fulfils condition (5). If f satisfies functional inequality (3), then there exist a Banach space E and additive mappings A : X → E and a : X → R such that f can be represented in the form: Proof. Clearly, f (0) = 0. In view of Lemma 1 and Theorem 1 of Ger the proof will be completed if we prove that each mapping g which solves (7) is sublinear. But this follows immediately from inequality (7) and from the homogeneity condition (5).
To ensure that each mapping which is of the form (8) solves (3) we need to show that if the group X appearing in Corollary 1 is a Hlawka space, then the space E postulated by Theorem 1 of Ger can be taken as a Hlawka space as well. Proof. The "if" part is obvious.
To prove the "only if" part let us apply Corollary 1 to derive that function f has the representation (8) with some Banach space E. Observe that H := A(X) is an additive subgroup of E. Therefore, to finish the proof we need to check the validity of (2) on H. For arbitrarily fixed x, y, z ∈ H let us pick u, v, w ∈ X such that We have as claimed. If in the foregoing theorem we assume additionally that the domain of f is a Banach space and moreover f is continuous, then we easily see that H is a linear subspace of E and additionally both mappings A and a are continuous.

Corollary 2.
Assume that X is a Banach space and f : X → R is continuous. Then f satisfies functional inequality (3) jointly with f (2x) = 2f (x) for all x ∈ X if and only if there exist a Hlawka space H, a continuous linear operator A : X → H and a continuous linear functional a : X → R such that f can be represented in the form (8).
If X and Y are arbitrary Abelian groups, then a mapping q : X → Y is called quadratic if and only if it satisfies the Jordan-von Neumann functional equation: for all x, y ∈ X. It is well known that if the group Y is uniquely divisible by 2, then for each quadratic mapping q : X → Y there exists a biadditive and symmetric mapping B : X × X → Y such that for all x ∈ X (see Aczél and Dhombres [1, Chapter 11, Proposition 1], compare also with Baron and Volkmann [4,Proposition]). Moreover, if additionally X is a Hilbert space, Y = R and function q is continuous, then B is a bilinear form and, consequently, there exists a continuous linear operator L : X → X such that q can be represented in the form In the next theorem we will provide a characterization of quadratic mappings via inequality (3).
if and only if f is quadratic. Moreover, if this is the case, then Eq. (4) is satisfied for all x, y, z ∈ X.
Proof. First we will prove the "only if" part. Using Lemma 1 and by our assumptions we easily see that the additive mapping a postulated by Lemma 1 vanishes. Next, from (7) we derive the inequality for all s, t ∈ X. Fix arbitrary u, v ∈ X and apply this estimate for s = u+v, t = u − v to obtain Fechner AEM for all u, v ∈ X. We reached two reverse inequalities and therefore f is quadratic. To verify the "if" part assume that f : X → R is a quadratic mapping and fix arbitrary x, y, z ∈ X. It is clear that (9) is satisfied. Let B : X × X → R be a biadditive and symmetric mapping such that Utilizing properties of B one can eventually transform (4) equivalently into an identity. Let us skip the straightforward calculation.

Corollary 3.
Assume that (X, +) is a Hilbert space and f : X → R is continuous. Then f satisfies functional inequality (3) jointly with (9) if and only if there exists a continuous linear operator L : X → X such that f can be represented in the form: We conclude this section with a description of solutions of inequality (3) under a more general homogeneity condition: It is clear that this condition is in particular fulfilled by every odd mapping satisfying (5) and by every even mapping satisfying (9). Proof. The "if" part is obvious.
To prove the "only if" part, let us define mappings a : X → R and q : X → R by the formulas By Lemma 1 we get that a is additive. Next, it is easy to see that condition (10) guarantees that q satisfies (9). Moreover, one can check that q fulfils inequality (3). Consequently, by Theorem 3 the mapping q is quadratic.
for all x ∈ X.

Inequality (3) on the real line
In what follows we will deal with solutions of functional inequality (3) on the real line with possibly weak additional assumptions (from now on no homogeneity is imposed upon f ). We will be considering the real line R equipped with the standard Lebesgue measure and we denote by R the set R ∪ {−∞, +∞}. Our main tool in this section are the Dini (extreme unilateral) derivatives.
For an arbitrary mapping f : R → R the Dini derivatives are defined as follows: It is clear that the Dini derivatives can attain infinite values. Therefore, later in this section each inequality which involves Dini derivatives is to be understood that it is valid provided that both its sides are meaningful (i.e. no indefinite expressions of the form ∞ − ∞ appears).
Banach proved that if f is measurable, then all Dini derivatives of f are measurable as well (see e.g. Saks [34,Chapter IV.4]).

Theorem 5 (Denjoy-Young-Saks). Assume that I is an interval and f : I → R
is an arbitrary function. Then there exists a set of measure zero C ⊂ I such that for all x ∈ I \ C exactly one of the following cases holds true: ( We will begin the study of functional inequality (3) on the real line with some lemmas. Lemma 2. Assume that f : R → R satisfies functional inequality (3) for all x, y, z ∈ R jointly with f (0) = 0. Then: for all x ∈ R.
Proof. We will prove (11) and (12) only. Proofs of (13) and (14) can be obtained by a modification of the original reasonings.

W. Fechner AEM
Fix arbitrary x ∈ R and y > 0 and apply (3) with substitution z = −x. We see that Next, let us pick a sequence (y n ) n∈N (possibly depending upon x) of positive real numbers which tend to zero and On replacing y n by its suitable subsequence we may assume additionally that both sequences are convergent in R. Next, apply (15) for y = y n , pass n → +∞ and estimate the two remaining limits by D + f (x) and 2D + f (0), respectively, to derive inequality (11). Further, for arbitrarily fixed x ∈ R and y > 0, apply (3) with substitution (x, y, z) → (x + y, −y, −x + y). We reach This time we choose a sequence (y n ) n∈N of positive real numbers tending to zero such that and the remaining limits are convergent. Applying an analogous reasoning as before for estimate (16) we prove inequality (12).
Proof. Let us keep x, z ∈ R temporarily fixed. For each y > 0 we deduce from (3) the following inequality: and its reverse for each y < 0.
Vol. 87 (2014) Hlawka's functional inequality 81 Now, pick a sequence (y n ) n∈N of positive real numbers which tend to zero such that By passing y → 0+ we obtain Analogously, in case y < 0 we can take a sequence (y n ) n∈N of positive real numbers tending to zero such that to get From estimates (17) and (18) we easily see that if the respective Dini derivative is finite at the origin, then the mapping D + f − D + f (0) is superadditive whereas the mapping D − f − D − f (0) is subadditive, respectively.
Remark 1. It is well known that for an arbitrary function f the equalities D ± f = +∞ and D ± f = −∞ can hold on a set which is at most countable (see e.g. Saks [34]). Therefore both subadditive mappings spoken of in the foregoing lemma are strictly greater than −∞ outside a countable set. Consequently, if we weaken assumptions of this lemma to: D + f (0) < +∞ or D − f (0) > −∞, then the assertion should be replaced by: there exists a countable set C ⊂ R such that the mapping −D + f + D + f (0) is well-defined on R\C and is subadditive on R\C, or then the mapping D − f − D − f (0) is well-defined on R\C and is subadditive on R\C, respectively. Now, we will recall a useful property of subadditive functions that can attain infinite values, which was first proved by Rosenbaum [33] (see also Hille and Phillips [16,Theorem 7.3.3]). Theorem 6 (Rosenbaum). If a subadditive measurable function ϕ : R → R satisfies ϕ(x 0 ) < +∞ for some x 0 < 0, then either ϕ(x) = +∞ for almost all x > 0 or ϕ < +∞ on R.
From this theorem we deduce the following useful fact. Proposition 1. Assume that ϕ : R → R is a subadditive measurable function. If ϕ(t) < +∞ for some t < 0 and for some t > 0, then either: