A Characterization of the Polarity Mapping for Convex Bodies

: In this paper, we establish a characterization of the polarity mapping for 1-dimensional convex bodies, which is a supplement to the result for such a characterization obtained by Böröczky and Schneider.


Introduction
Let R n denote the n-dimensional Euclidean vector space, equipped with its standard scalar product × × . We denote the set of convex bodies (compact convex subsets with nonempty interior) in R n which contain origin o in the interior by Κ n (o) . For K Î Κ n (o) , its dual or polar body K * is defined by (see, e.g., Ref. [1]) K * : = {x Î R n : xy ≤ 1 for all y Î K}.
Mahler s conjecture (see, e.g., Ref. [11]), a famous open problem, is related to polar bodies. By Κ n we denote the class of all convex bodies in R n , and by Κ n e we denote the class of all n-dimensional origin-symmetric convex bodies in R n . Let K Î Κ n e , the volume product of K and its polar body is defined by (see, e. g., Ref. [5]) P(K) = V (K)V (K * ), where V (K) denotes the n-dimensional volume of K. Along the volume product, there is the Mahler s conjecture that: for K Î Κ n e , P(K)≥ 4 n n! where equality holds for parallelepipeds and their polars (and other bodies). It is easily checked that P(K) = 4 for all K Î Κ 1 e . In 1939, Mahler [12] himself proved that P(K)≥ 8 for all K Î Κ 2 e , and in 1986, Reisner [13] characterized that equality holds only for parallelograms. Later, in 1991, Meyer [14] used some alternative methods to give a complete proof for the case n = 2, including the characterization of equality. Recently, Iriyeh and Shibata [15] showed that the conjecture holds for the case n = 3 and equality holds if and only if K or K * is a parallelepiped. For the case n ≥ 4, Mahler s conjecture is still a challenging open problem.
Duality for convex functions can also be defined. For a convex function f:R n ® R {±¥}, its conjugate function is defined by (see, e.g., Ref. [16]) If f is a lower semi-continuous convex function, then f * is also a lower semi-continuous convex function, and f ** = f. This duality for lower semi-continuous convex functions can be characterized from two simple and natural properties: involution and order-reversion. Artstein-Avidan and Milman [17] showed that any involution on the class of lower semi-continuous convex functions which is order-reversing, must be, up to linear terms, the well-known Legendre transform. For more results on the characterizations of the duality for convex functions, sconcave functions and log-concave functions, we can refer to Refs. [18][19][20][21][22].
Recently, Böröczky and Schneider [1] made use of an excellent tool that is lattice endomorphism from Gruber [23,24] , to characterize the duality mapping for convex bodies by interchanging the pairwise intersections and convex hulls of unions. Let Ú denote the convex hull of unions (see Sect.1 Notations for details).
Theorem 1 Let n ≥ 2 and let ϕ: for all KL Î Κ n (o) . Then either ϕ is constant, or there exists a linear transformation T Î GL(n) such that ϕ(K) = TK * for all K Î Κ n (o) . It is important to point out that the property (the duality interchanges the pairwise intersections and convex hulls of unions) is sufficient for a characterization, up to a trivial exception (the constant map) and the composition with a linear transformation. A mapping ϕ: for all K Î Κ n (o) . If ϕ satisfies condition (3) and one of conditions (1) and (2), then ϕ satisfies conditions (1), (2) and (3), and ϕ is order-reversing. By replacing condition (1) with (3), Böröczky and Schneider [1] completely established a characterization of the duality mapping for convex bodies in R n with n ≥ 2. Condition (3) excludes the constant map and forces the linear map appearing in the theorem to be selfadjoint.
Theorem 2 Let n ≥ 2 and let ϕ: for all KL Î Κ n (o) . Then there exists a selfadjoint linear . For more results on the characterization of duality and lattice endomorphism of the class of convex bodies and of convex sets, we can refer to Refs. [25][26][27][28][29].
The main purpose of this paper is to establish a characterization of the duality mapping for convex bodies on 1-dimensional Euclidean space with some additional assumptions. For simplicity, we will identify x Î(0 + ¥) with 0 < x < +¥ as follows. And, obviously, for all KL Î Κ n (o) and all real r > 0. Then, there exist constants cd Î R with cd > 0 such that for all xy Î(0 + ¥).

Notations
For reference, we collect some basic facts on convex sets and convex bodies. Excellent references are the books by Gardner [30] , Gruber [31] and Schneider [16] .
Let B stand for the unit ball {x Î  n : xx ≤ 1} and S n -1 the unit sphere of  n . A set A Ì  n is convex if for any two points xy Î A , the line segment [xy] joining them is contained in A, i.e., ( If AB are convex, then A + B ={x + y: for all x Î A y Î B} and rA ={rx: x Î A and r Î R} are convex. A convex body is a compact convex subset of R n with nonempty interior. The support function h K :R n ® R of a compact, convex K Ì R n is defined, for x Î R n , by h K (x) = max{ xy : y Î K} It can be easily checked that the support function is sublinear, i.e., h K has the positive homogeneity of degree 1 and satisfies subadditive. From the definition, it follows immediately that, for g Î GL(n), the support function of A boundary point x Î ¶K is said to have u Î S n -1 as an outer normal provided xu = h K (u). The convex body is equipped with the Hausdorff metric δ, which is defined for convex bodies KL by The set of convex bodies in R n containing the origin o in their interior is denoted by Κ n (o) . For K Î Κ n (o) , its dual or polar body K * is defined by The duality mapping K ® K * has a number of remarkable properties, of which we list the following; they are valid for all KL Î Κ n (o) : Continuity with respect to the Hausdorff metric; 6) If g Î GL(n), then (gK) * = g -t K * . If K È L is convex, then (K Ç L) * = K * È L * and (K È L) * = K * Ç L * . When n = 1, for KL Î Κ 1 (o) , K Ú L = K È L is convex. And for K Î Κ 1 (o) and real r ¹ 0, (rK) * = r -1 K * is the form of the case n = 1 in 6) above. For more interesting properties of the duality mapping of convex body, we can refer to Ref. [16], §1.6.

Main Results
for KL Î Κ 1 (o) . Note that if K Ì L, there is still ϕ(K) Ê ϕ(L). Then (6)  Without loss of generality, letting a > c, b < d. Then, we obtain from (4), (5) and (6) that We now study the first result f (cb) = max{ f (ab) f (cd)} which implies the following three possibilities, and the others are similar.
(P1) If f (ab)> f (cd), then we have f (cb) = f (ab). Thus, f (xy) is independent of the first variable x. We obtain that f (xy) can be rewritten as f (y), which is a decreasing positive function on (0 + ¥).
(P2) If f (ab)< f (cd), then we have f (cb) = f (cd). Thus, f (xy) is independent of the second variable y. We obtain that f (xy) can be rewritten as f (x), which is a decreasing positive function on (0 + ¥).
(P3) If f (ab) = f (cd), then we have f (cb) = f (ab) = f (cd). Thus, f (xy) is independent of the variable x and y. We obtain that f (xy) is a constant positive function on (0 + ¥). Since the decreasing function contains the constant function as a special case, f (xy) can be rewritten as either f (x) in (P2) or f (y) in (P1).
Consequently, we may rewrite f (xy) as either f (x) or f (y), and similarly, we rewrite g(xy) as either g(x) or g(y), where fg are two decreasing positive functions on (0 + ¥). Then, we conclude from (7) that ϕ([-xy]) has 4 different cases as desired.
Proof Let K Ì L, then, from (5), we obtain for KL Î Κ 1 (o) . Suppose ϕ is not order-reversing and we may assume that there exist two convex bodies which is a contradiction. Thus, (3) and (5) where f -1 ,g -1 denote the inverse function of f,g, respectively; or there exists a strictly decreasing positive func- where f -1 denotes the inverse function of f. Proof ϕ is order-reversing and satisfies (3), (4) and (5) via Lemma 2. Then, the functions f (xy), g(xy) must be strictly decreasing positive functions with respect to x and y on (0 + ¥). Thus, together with Lemma 1, we obtain that the functions f (xy), g(xy) be rewritten as f (x), f (y) and g(x),g(y), where f,g are two strictly decreasing positive functions on (0 + ¥) and that the forms of four cases in Lemma 1 are remained tentatively.
However, since ϕ is order-reversing, both Case 1 and Case 4 are removed when fg are two strictly decreasing positive functions on (0 + ¥). We show the contradiction for Case 1 (Case 4 is similar). Suppose y 1 < y 2 are arbitrary and x is fixed, then it follows from (8) (3)  for all λ > 0 and all K Î Κ 1 (o) . With the additional assumption of homogeneity, we establish a characterization of polarity or duality mapping for 1-dimensional convex