Dual mean Minkowski measures of symmetry for convex bodies

SCIENCE CHINA Mathematics. ARTICLES. July 016 Vol.59 No.7: 1383 1394 doi: 10.1007/s1145-016-511-x Dual mean Minkowski measures of symmetry for convex bodies GUO Qi 1, & TOTH Gabor 1 Department of Mathematics, Suzhou University of Science and Technology, Suzhou 15009, China; Department of Mathematics, Rutgers University, Camden, NJ 0810, USA Email: guoqi@mail.usts.edu.cn, gtoth@camden.rutgers.edu Received August 1, 015; accepted October 1, 015; published online January 15, 016 Abstract We introduce and study a sequence of geometric invariants for convex bodies in finite-dimensional spaces, which isin a sense dual to the sequence of mean Minkowski measures of symmetry proposed by the second author. It turns out that the sequence introduced in this paper shares many nice properties with the sequence of mean Minkowski measures, such as the sub-arithmeticity and the upper-additivity. More meaningfully, it is shown that this new sequence of geometric invariants, in contrast to the sequence of mean Minkowski measures which provides information on the shapes of lower dimensional sections of a convex body, provides information on the shapes of orthogonal projections of a convex body. The relations of these new invariants to the well-known Minkowski measure of asymmetry and their further applications are discussed as well. Keywords geometric invariant, measure of symmetry, dual measure of symmetry, simplex, affine diameter MSC(010) 5A0, 46B99, 5A38, 5A39, 5A40 Citation: Guo Q, Toth G. Dual mean Minkowski measures of symmetry for convex bodies. Sci China Math, 016, 59: 1383 1394, doi: 10.1007/s1145-016-511-x 1 Introduction As one of the most important geometric invariants, the measure of symmetry (or asymmetry) of convex bodies(i.e., compactconvexsetswith nonemptyinteriorinr n, thestandardeuclideanspace), formulated by Grünbaum in his well-known paper [3], has regained much attention in recent years (see [1, 4 6, 9 11, 13,16] and the references therein). So, some new measures have been found (see [1,5,11,13,16,1]), and more properties of the known ones, including the stability and the relations with other kinds of geometric invariants, are revealed (see [1 3, 6, 14, 18, 19, ]), and as consequences, some new geometric inequalities are established (see [1,,4,5,9 11,14,]). In general, measures of symmetry (or asymmetry) can be used in geometry to measure how far a convex body (as a whole) is from some particular convex bodies, e.g., centrally symmetric convex bodies, convex cones or simplices. However, meaningfully, Toth[16] introduced a family of measures(functions) of symmetry σ m (see below for definition), m 1, called the mean Minkowskimeasures of symmetry, which, prior to most measures of symmetry (or asymmetry), measure not only convex bodies themselves but also their lower-dimensional sections. Roughly speaking, for a convex body K, its (m-th) mean Minkowski measure of symmetry σ m is a function defined on intk, the interior of K, which, when 1 < m n, provides information on the shapes of m-dimensional sections of K. Corresponding author c Science China Press and Springer-Verlag Berlin Heidelberg 016 math.scichina.com link.springer.com

1384 Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 The properties and applications of the mean Minkowski measures of symmetry have been investigated in a series of papers (see [16 ]), where, as an application of the mean Minkowski measures, readers may find in particular a partial answer to the long-standing Grünbaum conjecture for the existence of n+1 affine diameters meeting at one point of a convex body (see [3]). In this paper, we introduce another family of measures (functions) of symmetry σm, m 1, called the dual mean Minkowski measures of symmetry, which in a sense are dual to the mean Minkowski measures. It turns out that dual mean Minkowski measures share almost all nice properties with mean Minkowski measures and, in sharp contrast to the mean Minkowski measures, describe the shapes of orthogonal projections of a convex body. Furthermore, the dual mean Minkowski measures are relatively easier in computation than the mean Minkowski measures, and can also be applied to deal with the Grünbaum conjecture mentioned above as well (see [7]). Notation and definitions Let K n denote the family of all convex bodies in R n. For any subset S R n, convs and cones denote the convex hull and the convex conical hull of S, respectively. lins denotes the linear subspace generated by S. A map T : R n R m is called affine if T(λx+(1 λ)y) = λt(x)+(1 λ)t(y) for any x,y R n and λ R. In particular, an affine map f : R n R is called an affine function. It is known that f : R n R is affine if and only if f( ) = u, +b for some unique u R n and b R, where, denotes the classical inner product. Denote by aff(r n ) the family of affine functions on R n and by Aff(R n ) the family of affine maps from R n to R n. We refer the readers to [15] for other notation and terms. The (n 1)-dimensional unit sphere is denoted by S n 1. An n-dimensional simplex (n-simplex for brevity) is denoted by n, i.e., n := conv{v 0,v 1,...,v n }, where v 0,v 1,...,v n R n, called the vertices of n, are affinely independent. We recall the well-known Minkowski measure of asymmetry: Given a convex body K K n and x intk, for a hyperplane H through x and the pair of support hyperplanes H 1,H of C parallel to H, let r(h,x) be the ratio, not less than 1, in which H divides the distance between H 1 and H. Note as (x) = as (K,x) := max{r(h,x) H x}, and the Minkowski measure as (K) of asymmetry of K is defined by (see [3,1]) as (K) = min x int(k) as (K,x). A point x int(k) such that as (K,x) = as (K) is called a Minkowski (or -) critical point (of K). The set of all -critical points of K is denoted by C. Another equivalent definition is as follows: Let l := pq be a chord of K passing through x, where p,q bdk, the boundary of K. If defining d(p,x) γ(k,x) := max l x d(x,q), where d(, ) is the Euclidean metric, then γ(k,x) = as (K,x) (see [1]) and so as (K) = min x int(k) γ(k,x). It is known that for any K K n, 1 as (K) n, and as (K) = 1 iff K is (centrally) symmetric and as (K) = n iff K is an n-dimensional simplex (see [3,1]). Next, we recall the definition of the mean Minkowski measure (function) of symmetry introduced in [16]. Some notation is needed first. Let K K n and x intk. A multi-set {c 0,c 1,...,c m } (m 1 and repetitions are allowed), where c 0,c 1,...,c m bdk, is called an m-configuration of K with respect to (w.r.t. for brevity) x if x conv{c 0,c 1,...,c m }. Denote by C m (x) = C K,m (x) the family of m-configurations of K w.r.t. x. Definition.1 (See [16]). Given K K n, for each m 1, we define its (m-th) mean measure (function) of symmetry σ m = σ K,m : intk R by σ m (x) := inf m {c 0,...,c m} C m(x) 1 Λ(c i,x)+1, x intk,

Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 1385 where Λ(c,x) = Λ K (c,x) := d(c,x) d(c o,x) is the distortion and co bdk denotes the opposite point of c bdk against x. Clearly, for each m 1, σ m is affinely invariant. The following theorem was proved in [17]. Theorem. (See [17]). Let K K n. For m 1, we have 1 σ m m+1. If m, then σ m (x) = (m+)/ for some x intk iff K is symmetric with respect to x. If σ m (x) = 1 for some x intk, then m n and K has an m-dimensional simplicial intersection across x, i.e., there is an m-dimensional hyperplane H such that x H and K H is an m-simplex. Conversely, if K has a simplicial intersection with an m-dimensional hyperplane H, then σ m = 1 identically on K H. From Theorem., we see that the mean Minkowski measures provide indeed information about the lower dimensional sections of a convex body. Now we introduce a dual measure to the mean Minkowski measure, called the dual mean Minkowski measure. In order to do so, we need some more notation. Given K K n, we define the set K a [0,1] by K a [0,1] := {f aff(rn ) f(k) = [0,1]}. It is easy to see that if f K[0,1] a, then {f = 0} and {f = 1} are a pair of (parallel) support hyperplanes of K, from which it follows that as (K,x) = max{ 1 f(x) f(x) f K[0,1] a },x intk (see [4,5]). Given K K n, for each m 1, we define its m-support configuration in the following way: A multi-set {f 0,...,f m } K[0,1] a (repetitions are allowed) is called an m-support configuration of K if m {f i 0} =, where {f i 0} := {x R n f i (x) 0}. The family of m-support configurations of K is denoted by C m = C K,m. Remark.3. In contrast to C m (x), C m does not depend on any point in the interior of K. Definition.4. Let K K n. For each m 1, its (m-th) dual mean Minkowski measure (function) σm = σ K,m : intk R is defined by { m σm (x) := inf } f i (x) {f 0,...,f m } Cm, x intk. A point x intk satisfying σ m(x ) = sup x intk σ m(x) is called a σ m-critical point of K. Remark.5. (1) The dual mean Minkowski measure is indeed a dual concept of the mean Minkowski measure (see Corollary 3.5). () Each σm is concave in intk since it is the infimum of some concave functions m f i(x), whereas σ m is not concave in general (see [19]). Thus σm and σ m do not coincide in general. (3) Since σm is concave, σm(x) 1, x intk (see Corollary 3.11) and lim x bdk σm(x) = 1 (see Proposition 4.4), there exists at least one σm -critical point. (4) σ1 1 trivially in intk since {f 0,f 1 } C1 iff f 1 = 1 f 0. Among other conclusions, one of the main results in this paper is the following theorem. Theorem.6. Let K K n. For m 1, we have 1 σ m m+1. If m, then σ m(x) = (m+1) for some x intk iff K is a symmetric body centered at x. If σ m(x) = 1 for some x intk, then m n, σ m 1 and K has an m-dimensional simplicial projection, i.e., there is a projection P H : R n H, where H is an m-dimensional subspace, such that P H (K) is an m-simplex. Conversely, if K has an m-dimensional simplicial projection ( m n), then σ m 1. From Theorem.6, we see that the dual mean Minkowski measures provide indeed information about the lower-dimensional orthogonal projections of a convex body. The paper is organized as follows: Section 3 discusses the characteristics and properties of the support configurations. Section 4 studies the basic properties, such as sub-arithmeticity and upper-additivity etc., of the dual mean Minkowski measure sequences. Section 5 is devoted to the proof of Theorem.6. Finally, Section 6 presents the conclusions and further considerations.

1386 Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 3 Properties of support configurations In this section, we discuss the properties of support configurations and show some of their characteristics. We first prove the following theorem. Theorem 3.1. Let f i (x) K[0,1] a,i = 0,1,...,m, where f i( ) = u i, + b i. Then, the following statements are equivalent: (1) {f i } m C m. () For each u R n, u i,u 0 for some i. (3) o ri(conv{u i0,...,u il }) for some affinely independent u i0,...,u il, 1 l min{m,n}, i.e., there are positive α ik such that l α i k u ik = o. (4) cone{u i0,u i1,...,u il } = lin{u i0,u i1,...,u il } for some affinely independent u i0,u i1,...,u is, 1 l min{m,n}. In order to prove Theorem 3.1, we need the following lemma. Lemma 3.. Let f i (x) K a [0,1],i = 0,1,...,m, where f i( ) = u i, +b i. Then, for any (u m+1,b m+1 ) cone{(u 0,b 0 ),...,(u m,b m )}, we have where f m+1 ( ) = u m+1, +b m+1. m m+1 {f i 0} = {f i 0}, Proof. If m {f i 0} =, then the equality is obvious. If m {f i 0}, then this is just a reformulation of the well-known generalized Farkas lemma (see [8, p. 60]). Proof of Theorem 3.1. (1) () Let {f i } m C m and u R n. If u i,u > 0 or u i, u < 0 for all i, then f i ( λu) = u i, λu +b i as λ + for all i. Thus, λu m {f i 0} for sufficiently large λ, a contradiction to (1). () (3) Consider the linear subspace V := cone{u i } m ( cone{u i} m ). If V = {o}, then by the separation theorem for cones there is u R n such that {u i } m {x u,x > 0}, which contradicts (). So dimv 1. Now, it is easy to see by the definition that V = cone{u i0,...,u is } = lin{u 0,...,u s } for some u i0,...,u is (1 s m). Thus, since u i0 lin{u ik } s = cone{u i k } s, we have u i 0 = α 0 u i 0 + s k=1 α i k u ik with α 0,α i k 0. Thus o = s α i k u ik, where α i0 := 1+α 0 > 0. Now, let l be the smallest positive integer such that o = l α i k u ik for some u i0,...,u il with α ik > 0. Clearly l 1. We claim that u i0,...,u il are affinely independent. Suppose u i0,...,u il are not affinely dependent, then l β i k u ik = o for some (not all zero) β i0,...,β il with l β i k = 0. Let { } αik λ := min β i k < 0 = (say) α i l, β il then o = l α ik u ik + β ik l λβ ik u ik = l l 1 (α ik +λβ ik )u ik = (α ik +λβ ik )u ik, where α ik +λβ ik 0 (0 k l 1) and at least one of them is positive, a contradiction to the choice of l. Thus, o = 1 l α i k l α ik u ik = l α ik l α i k u ik ri(conv{u i0,...,u il }), where l min{m,n} clearly. (3) (4) Without loss of generality, suppose that u 0,...,u l, 1 l min{m,n}, are affinely independent and o ri(conv{u 0,...,u l }), i.e., o = l α iu i with α i > 0. Then for each 0 i l, we have u i = l α j j i α i u j cone{u 0,...,u l }. Thus lin{u 0,...,u l } = cone{u 0,...,u l }.

Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 1387 (4) (1) Without loss of generality, suppose cone{u 0,...,u l } = lin{u 0,...,u l } with affinely independent u 0,...,u l for some 1 l min{m,n}, then u 0 = l α iu i, where α i 0 with at least one α i positive. Thus, setting f m+1 ( ) := u 0, + l α ib i, we have, by Lemma 3.1, m m+1 {f i 0} = {f i 0} = since {f 0 0} {f m+1 0} = (observing that inf f m+1(x) = inf x intk x intk = t t α i u i,x + α i inf x intk u i,x + t α i b i t α i b i t ( ) α i inf u i,x +b i = 0 x intk and so {f m+1 0} {f 0 1}). Therefore {f 0,f 1,...,f m } C m. We now present some other basic properties of support configurations. Proposition 3.3. Let K K n, f 0,...,f m K[0,1] a,m 1. (1) {f 0,...,f m } Cm iff {1 f 0,...,1 f m } Cm. () m {f i 0} = iff m {f i < 0} =. (3) Cm is compact in R(n+1)(m+1). Proof. (1) Write f i ( ) = u i, +b i, then 1 f i (x) = u i,x b i +1,i = 0,1,...,m. Thus {f 0,...,f m } Cm, by Theorem 3.1(), iff for each u R n, u,u i 0 for some 0 i m, i.e., iff for each u R n, u, u i 0 for some 0 i m, and so, by Theorem 3.1 again, iff {1 f 0,...,1 f m } Cm. () m {f i 0} = implies clearly m {f i < 0} =. If m {f i 0}, choosing x 1 m {f i 0} and x intk (so f i (x ) > 0 for all i), we have, since f i (λx 1 +(1 λ)x ) = λf i (x 1 )+(1 λ)f i (x ) as λ + for each i, f i (λ 0 x 1 +(1 λ 0 )x ) < 0 for some λ 0 > 0 and all i. Thus, λ 0 x 1 +(1 λ 0 )x m {f i < 0}, i.e., m {f i < 0}. (3) Since K[0,1] a is compact in Rn+1 (see [9, Lemma 1]) and so is K[0,1] a Ka [0,1] ((m+1)-fold), we need only to show that Cm Ka [0,1] Ka [0,1] is closed. Let {f (k) 0,...,f m (k) } Cm,k = 1,,..., and {f (k) 0,...,f m (k) } {f 0,...,f m } K[0,1] a Ka [0,1] as k, which is equivalent to f (k) i f i for each i. Now, suppose {f 0,...,f m } / Cm, i.e., m {x f i(x) 0}, then by () just proved above, there exists x 0 m {x f i(x) < 0}, i.e., f i (x 0 ) < 0 for each i. Thus, since f (k) i (x 0 ) f i (x 0 ) as k, we conclude that there is k 0 such that f (k) i (x 0 ) 0 for all k k 0 and all i, which contradicts that {f (k) 0,...,f m (k) } Cm, k k 0. Corollary 3.4. The infimum in the definition of σm is attainable, i.e., for given x intk, there is an m-support configuration {f 0,...,f m } CK,m such that m f i(x) = σm (x). Proof. This follows from Proposition 3.3(3) and the fact that m f i(x) is continuous w.r.t. {f 0,...,f m }. An m-support configuration {f i } such that m f i(x) = σm(x) is called an m-minimizer w.r.t. x intk. The next corollary shows a kind of duality between σ m and σm. Before stating the corollary, we need some preparations. Given K K n and x intk, we define the support function h x ( ) = h x (K, ) based at x of K by h x (K,u) := sup{ u,y x y K}, u R n,

1388 Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 the gauge function g x ( ) = g x (K, ) based at x of K by and the dual body K x based at x of K by g x (K,u) := inf{λ 0 u λ(k x)}, u R n, K x := {y y,z x 1,z K}+x. Clearly, when x = o intk, h o,g o and K o are exactly the classical ones, and h x (K,u) = h o (K x,u), g x (K,u) = g o (K x,u), K x := (K x) o +x, h x (K x,u) = h o ((K x) o,u), g x (K x,u) = g o ((K x) o,u). [15, Lemma 1.7.13]states an elegant relation between the support and the gauge function: If o intk, then for any u R n, g o (K,u) = h o (K o,u). Corollary 3.5. Let K K n and m 1. Then, σ K,m (x) = σ K x,m (x), σ K,m (x) = σ K x,m(x). Proof. We first point out a fact that for each u R n \ {o}, there are (unique) µ > 0 and b R such that f( ) := µu, + b K[0,1] a (choosing b = b 1/a and µ = 1/a, where b 1 := inf x K u,x and a := sup x K u,x +b 1 ). Then, for c i bdk, defining f i ( ) := µ i (c i x), + b i (K x ) a [0,1], 0 i m, where µ i > 0 as mentioned above, we have {c 0,c 1,...,c m } C K,m (x) x conv{c 0,c 1,...,c m } (by definition) l x = α ik c ik, where l 1,{c ik } {c i },α ik > 0, o = l l α ik = 1 (by x / bdk) α ik µ ik µ ik (c ik x) {f 0,f 1,...,f m } C K x,m (by Theorem 3.1). Next, observing that for c bdk, d(x,ci) d(x,c i ) = g o(k x,x c) and g o (K x,c x) = 1 since c x,c x bd(k x), we obtain 1 Λ(c i,x)+1 = 1 d(x,c i) d(x,c i ) +1 = g o (K x,c i x) g o (K x,x c i )+g o (K x,c i x) h o ((K x),c i x) = h o ((K x),x c i )+h o ((K x) (by [15, Lemma 1.7.13]),c i x) h x (K x,c i x) = h x (K x,x c i )+h x (K x,c i x) = 1 f i(x). Hence, with the help of Proposition 3.3, we have σ K,m (x) = inf m {c 0,c 1,...,c m} C K,m(x) = inf {f 0,f 1,...,f m} C K x,m 1 Λ(c i,x)+1 m (1 f i (x)) = The second equality follows from the fact that (K x ) x = K. inf {f 0,f 1,...,f m} C K x,m m f i (x) = σk x,m (x). One kind of particular support configurations will play an important role in the study of dual mean Minkowski measures of symmetry.

Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 1389 Definition 3.6. Let 1 m n. For 1 m n, an {f 0,...,f m } Cm, where f i( ) = u i, +b i, is called simplicial if m {f i 0} lin{u i } m is an m-simplex, where a 1-simplex means a segment. Theorem 3.7. Let {f 0,...,f m } Cm with f i ( ) = u i, + b i (1 m n). Then the following are equivalent: (1) {f 0,...,f m } is simplicial. () u 0,...,u m are affinely independent and cone{u i } m = lin{u i} m. (3) m α iu i = o for some positive α 0,...,α m and if m i=1 β iu i = o with non-negative β i, then either β i > 0 for all i or β i = 0 for all i. (4) {f 0,...,f m } has no proper sub-support configurations. Proof. (1) () If {f 0,...,f m } is simplicial, then u 0,...,u m are exactly the m + 1 inner normals of facets of the m-simplex m := m {f i 0} lin{u i } m. So {u i} m are affinely independent and dim(lin{u i } m ) = m. Suppose that cone{u i} m lin{u i} m. Then by the Separation Theorem for cones, cone{u i } m {x lin{u i} m u,x 0} for some u lin{u i} m. Thus, choosing x 0 m, we have f i (x 0 +tu) = u i,x 0 +tu +b i = u i,x 0 +b i +t u i,u 0 for all t 0 and all i, i.e., {x 0 +tu t 0} m, which contradicts the boundedness of m in lin{u i } m (noticing that x 0 +tu lin{u i } m for all t). () (3) Since cone{u i } m = lin{u i} m, we have u 0 = m γ iu i for some γ i 0. Thus m α iu i = o, where α 0 = 1+γ 0 > 0 and α i = γ i 0 for 1 i m. Before showing that all α i above are positive, we first show the second conclusion. Assume that m β iu i = o forsomenon-negativeβ 0,...,β m. Suppose thatsomeβ i arezerosandsomeβ i arepositive. Without loss of generality, we assume β m = 0 (and so m 1 β i o). Observing that u m = m β i u i or equivalently m 1 β i u i +(1+β m )u m = o for some β i 0, we have m 1 (µβ i β i )u i (1+β m )u m = o and m 1 (µβ i β i ) (1+β m ) = 0, where µ = ( m 1 β i) 1 ( m 1 β i +(1+β m )), which contradicts the affine independence of u 0,...,u m since 1+β m 0. Now the fact that all α i are positive is just a simple consequence of the second conclusion. (3) (4) Without loss of generality, suppose {f 0,...,f m1 } C m 1, where 1 m 1 < m, then by Theorem 3.1(3), there are affinely independent u i0,...,u il, 1 l min{m 1,n} < m, such that o ri(conv{u ik } l ), i.e., l k=1 α i k u ik = o with α ik > 0, which contradicts (3) since l < m. (4) (1) By Theorem 3.1, o ri(conv{u i0,...,u il }) for some affinely independent u i0,...,u il, 1 l m, which in turn shows {f i0,...,f il } C l by Theorem 3.1 again. Thus l = m by (4) and so o = m α iu i with all α i > 0. Now we claim that the non-empty set m := m {f i 0} lin{u i } m is bounded in the m-dimensional subspace lin{u i } m which means that {f 0,...,f m } is simplicial. Suppose that m is not bounded in lin{u i } m, then {x 0 +tu t 0} m for some x 0 m and non-zero u lin{u i } m (so f i(x 0 +tu) 0 for all t > 0 and all i). However, since 0 = m α i u i,u, we have u i0,u < 0 for some i 0 and furthermore a contradiction. 0 f i0 (x 0 +tu) = u i0,x 0 +t u i0,u +b i0 as t +, The next proposition shows that the simplicial support configurations are not very special. Proposition 3.8. Any {f 0,...,f m } Cm (m 1) has a simplicial sub-support configuration {f i0,...,f il } Cl, where 1 l min{m,n}. Proof. Denote f i ( ) = u i, +b i. Then, let l be the smallest positive integer such that l α i k u ik = o for some {u ik } l {u i} m and α i k > 0, 0 k l (by Theorem 3.1, such l exists and 1 l min{m,n}). Thus, {f i0,...,f il } Cl by Theorem 3.1 again, and further {f i0,...,f il } is l-simplicial by Theorem 3.7(3).

1390 Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 We end this section with the following proposition and its corollary. Proposition 3.9. Let K K n. If {f 0,...,f n } CK,n is n-simplicial, then n f i(x) 1 for all x intk and the equality holds for some x intk iff K = n {x f i(x) 0}, i.e., K is an n-simplex. In order to prove Proposition 3.9, we need the following well-known fact and we repeat the proof here for completeness (see [4]). Lemma 3.10. Let K n be an n-simplex with vertices v 0,...,v n and g i a [0,1] be such that g i (v j ) = δ ij, the Kronecker symbol. Then n g i 1 in R n. Proof. have, Observing that for any x R n, x = n α iv i for some α 0,...,α n R with n α i = 1, we n g i (x) = n ( n ) g i α j v j = j=0 n j=0 n α j g i (v j ) = n α i = 1. Proof of Proposition 3.9. Denote := n {x f i(x) 0} which is an n-simplex with vertices, say, v 0,...,v n. Let g i a [0,1], 0 i n, be such that g i(v j ) = δ ij. Then it is easyto checkthat g i (x) f i (x) for all x K and all i since K. Thus, we obtain n f i(x) n g i(x) = 1 by Lemma 3.10. For the equality case, n f i(x) = 1 for x intk iff g i (x) = f i (x) and iff g i = f i for all i and so iff K =. Corollary 3.11. Let K K n and m 1. Then σm (x) 1 for all x intk. Proof. For any x intk, let {f 0,...,f n } CK,n, where f i( ) = u i, +b i, be a minimizer w.r.t. x, i.e., m f i(x) = σm (x). By Theorem 3.7, there is a simplicial l-support configuration {f i 0,...,f il } {f 0,...,f n } (1 l min{m,n}). Therefore, with the help of Proposition 3.9, we obtain that, for any x intk, m l l σm(x) = f i (x) f ik (x) = f ik (Px) 1, where P is the orthogonal project from R n to lin{u i0,...,u il }. 4 Properties of the sequence {σ m} m 1 In this section, we show that the dual measures σm share many nice properties with the mean Minkowski measures. First we show that, similar to {σ m } m 1, the sequence {σm } is sub-arithmetic. Theorem 4.1. For any K K n,x intk and m,k 1, we have σ m+k (x) σ m (x)+k ˆf(x), x intk, where ˆf = ˆf x K[0,1] a satisfies ˆf(x) 1 ˆf(x) = inf f K a f(x), or equivalently = as [0,1] ˆf(x) (K,x). Moreover, the equality holds for m = n and k 1, i.e., the sequence {σn+k } k 1 is arithmetic. Proof. For any {f 0,...,f m } Cm, denote by {f 0,...,f m, ˆf,..., ˆf} the (m+k)-support configuration with k copies of ˆf added to {f 0,...,f m }. Then we have σ m (x)+k ˆf(x) = inf {f 0,...,f m} C m m f i (x)+kˆf(x) = inf {f 0,...,f m,ˆf,...,ˆf} C m+k m+k inf {f 0,...,f m+k } Cm+k ( m ) f i (x)+kˆf(x) f i (x) = σm+k(x).

Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 1391 If k 1, then, for any {f 0,...,f n+k } C n+k, by Helly s theorem we have, say, {f 0,...,f n } C n. Thus, for x intk, n+k f i (x) = n f i (x)+ m+k i=n+1 σ n(x)+k ˆf(x), which leads clearly to σ n+k (x) σ n (x) + k ˆf(x) and in turn (together with the reverse inequality just proved above) leads to σ n+k (x) = σ n(x)+k ˆf(x), i.e., the sequence {σ n+k } k 1 is arithmetic. f i (x) We now show that, similar to {σ m }, the sequence {σm } is also upper-additive. Theorem 4.. Let K K and m,k 1. Then we have σ m+k σ m+1 σ k σ 1. Proof. Let {f 0,...,f m+k } Cm+k be an (m+k)-minimizer w.r.t. x intk. If {f 0,...,f m } Cm, then by applying Theorem 4.1 repeatedly, we have m σm+k (x) = f i (x)+ m+k i=m+1 σm (x)+k ˆf(x) f i (x) σ m+1(x)+(k 1)ˆf(x) = σ m+1 (x)+(σ 1 (x)+(k 1)ˆf(x)) σ 1 (x) σ m+1(x)+σ k(x) σ 1(x), where ˆf is the same as in Theorem 4.1. If {f m+1,...,f m+k } Ck 1, we have, by applying Theorem 4.1 repeatedly again, m σm+k (x) = f i (x)+ m+k i=m+1 f i (x) (m+1)ˆf(x)+σ k 1 (x) = (mˆf(x)+σ 1(x))+(ˆf(x)+σ k 1(x)) σ 1(x) σ m+1 (x)+σ k (x) σ 1 (x). Now, suppose that {f 0,...,f m } / Cm and {f m+1,...,f m+k } / Ck 1. We first observe that, by Theorem 3.1, there are non-negative α i, i = 0,1,...,m + k, with α i0 > 0 for at least one i 0 such that m+k α iu i = o, from which we get m α iu i = m+k i=m+1 α iu i =: u. We claim that u o (so not all α i, i = 0,...,m, are zero and not all α i, i = m+1,...,m+k, are zero) for otherwise we would have by Theorem 3.1 again {f 0,...,f m } Cm or {f m+1,...,f m+k } Ck 1. We set f( ) := u, +b K[0,1] a (and so 1 f = u, +1 b Ka [0,1] as well), where b = m α ib i. Thus, it is easy to check that by Lemma 3. m {f i 0} {1 f 0} = m {f i 0} =, i.e., {f 0,...,f m,1 f} Cm+1. Similarly, {f m+1,...,f m+k,f} Ck. Thus, σ m+k(x) = where we used the fact σ 1 1. m f i (x)+ m+k i=m+1 f i (x) ( m ) = f i (x)+(1 f(x)) σ m+1(x)+σ k(x) σ 1(x), ( m+k + i=m+1 ) f i (x)+f(x) 1

139 Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 The following are some applications of Theorem 4.1: The first one reveals the relation between the Minkowski measure of asymmetry and the dual mean Minkowski measure of symmetry and the second one implies that the concave function σ m : intk R can be continuously extended to the whole K by setting it equal to 1 on bdk. Proposition 4.3. Let K K n. Then lim m σ m (x) m = 1 1+as (x). Proof. By Theorem 4.1 and the fact that σ m(x) (m+1)ˆf(x), where ˆf is the same as in Theorem 4.1, we have m+1 ˆf(x) = lim m m ˆf(x) lim m lim m So lim m σ m (x) m = ˆf(x) = 1 1+as (x). σ 1 (x)+(m 1)ˆf(x) m σ m (x) m = lim m = ˆf(x). σ 1+(m 1) (x) Proposition 4.4. Let K K n and x 0 bdk. Then lim x x0 σm (x) = 1. Proof. By Corollary 3.11 and Theorem 4.1, we have 1 lim x x 0 σ m(x) lim x x 0 (σ 1(x)+(m 1)ˆf x (x)) = lim x x 0 (1+(m 1)ˆf x (x)) = 1, where we used the obvious facts that σ 1 1 and lim x x0 ˆfx (x) = 0. Therefore lim x x0 σ m(x) = 1. m 5 Proof of the main theorem We start this section with two lemmas which will be needed for the proof of Theorem.6. Lemma 5.1. If σn(x) = 1 for some x intk and there is a simplicial n-minimizer w.r.t. x, then K is an n-simplex (and so σn 1). Proof. Suppose σn(x) = 1 for some x intk and {f 0,...,f n } Cn is a simplicial minimizer w.r.t. x, i.e., := n {f i 0} is an n-simplex. For 0 i n, let g i a [0,1] be such that {g i = 0} = {f i = 0}, then g i (x) f i (x) (since K ). If K is not an n-simplex, then there exists at least one i such that g i (x) < f i (x). Thus σ n (x) = n f i(x) > n g i(x) = 1 (where the last equality follows from Lemma 3.10), a contradiction. Lemma 5.. If σm (x) = 1 (m n) for some x intk, then all m-minimizers w.r.t. x are simplicial. Proof. If {f 0,...,f m } C m is not m-simplicial, then by Theorem 3.7 and Proposition 3.8, {f 0,...,f m } has a proper sub-support configuration, say, {f 0,...,f l }, where l < m. Thus we have m f i (x) = l f i (x)+ So {f 0,...,f m } is not an m-minimizer w.r.t. x. Now, it is the time to prove Theorem.6. m i=l+1 f i (x) > σl (x) 1. Proof of Theorem.6. First, we consider the inequalities. Since σm 1 was already shown in Corollary 3.11, we need only to show σm m+1. By Theorem 4.1, σ m(x) = σ 1+(m 1) (x) σ 1(x)+(m 1)ˆf(x) 1+ m 1 = m+1, where we used the fact that ˆf(x) 1 for all x intk (which can be easily seen from the definition of ˆf).

Guo Q et al. Sci China Math July 016 Vol. 59 No. 7 1393 We now discuss the equality cases. Suppose that K is a symmetric body centered at x. Then f(x) = 1 for all f Ka [0,1] and so m f i(x) = m+1 for any {f 0,...,f m } Cm, and in turn σm(x) = m+1. Conversely, if σm m+1 (x) = for some x intk, then by Theorem 4.1, m+1 = σm (x) σ m 1 (x)+ ˆf(x) m + 1 = m+1, where we used the fact that σm 1 (x) m and ˆf(x) 1, which implies clearly ˆf(x) = 1. Thus, by the definition of ˆf, we have f(x) = 1 for all f Ka [0,1] and in turn that K is a symmetric body centered at x. Now suppose there is an orthogonal project P H : R n H, an m-dimensional subspace ( m n), such that := P H (K) is an m-simplex in H with vertices, say, v 0,...,v m. Let affine functions f i : H R (0 i m) be such that f i (v j ) = δ ij. Then { f 0, f 1,..., f m } C,m and m f i 1 in int. Now, setting f i := f i P H : R n R (0 i m), we have clearly {f 0,...,f m } CK,m and m f i(x) = m f i (P H x) = 1 for x intk, i.e., σm 1. Conversely, if σm(x) = 1 for some x intk and {f 0,...,f m } Cm is an m-minimizer w.r.t. x, i.e., m f i(x) = 1, then m n for otherwise, by Helly s theorem there are, say, f 0,...,f n such that {f 0,...,f n } Cn, and so we would have 1 = σm(x) = m f i(x) = n f i(x) + m i=n+1 f i(x) σn (x)+ m i=n+1 f i(x) > 1. By Lemma 5., {f 0,...,f m } is m-simplicial. Now setting H := lin{u i } m where u 0,...,u m are such that f i ( ) = u i, +b i, we have, by the definition of simplicial support configurations, that P H (K) is an m-simplex in H. The proof is complete. 6 Conclusions and further considerations As mentioned in Introduction and shown in later sections, in contrast to the mean Minkowski measures which describe the shapes of low-dimensional sections of a convex body, the dual mean Minkowski measures of symmetry provide indeed some valuable information on low-dimensional orthogonal projections of a convex body, in particular, on low-dimensional simplicial projections. To the best of our knowledge, the mean Minkowski and the dual mean Minkowski measures of symmetry are probably the only measures which provide information not only on the shape of a convex body itself but also on the shapes of its sections or projections. We expect more such kinds of measures of symmetry (or asymmetry) to be found. In this paper, we pay attention mainly to the best lower/upper bounds of dual mean Minkowski measures and determining the corresponding extremal projections. There are still more problems to be studied, e.g., the stabilities of dual mean Minkowski measures at both the extremal values, i.e., 1 and m+1 ; the properties of the set of σ m-critical points. Also, we think it hopeful that Grünbaum s conjecture is valid at a σn-critical point, i.e., there should be n+1 affine diameters meeting at one σn-critical point of a convex body (see [7]). Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 11718) and the Jiangsu Specified Fund for Foreigner Scholars 014 015. The authors thank the referees for their carefully reading the manuscript and pointing out some typos. References 1 Belloni A, Freund R M. On the asymmetry function of a convex set. Math Program Ser B, 008, 111: 57 93 Groemer H. Stability theorems for two measures of symmetry. Discrete Comput Geom, 000, 4: 301 311 3 Grünbaum B. Measures of symmetry for convex sets. In: Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, vol. 7. Providence: Amer Math Soc, 1963, 33 70 4 Guo Q. Stability of the Minkowski measure of asymmetry for convex bodies. Discrete Comput Geom, 005, 34: 351 36

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