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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


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Volume 19, Issue 1

Issues

Is a complete, reduced set necessarily of constant width?

René Brandenberg / Bernardo González Merino / Thomas Jahn / Horst Martini
Published Online: 2019-01-18 | DOI: https://doi.org/10.1515/advgeom-2017-0058

Abstract

Is it true that a convex body K being complete and reduced with respect to some gauge body C is necessarily of constant width, i.e. does it satisfy KK = ρ(CC) for some ρ > 0? We prove this implication for several cases including the following: if K is a simplex or if K contains a smooth extreme point, then the implication holds. Moreover, we derive several new results on perfect norms.

Keywords: Bodies of constant width; Bohnenblust’s inequality; complete bodies, gauges; generalized Minkowski spaces; Leichtweiss’ inequality; normed spaces; perfect norms; reduced bodies

MSC 2010: 46B20; 52A20; 52A21; 52A40; 52B11

1 Introduction

The notions of constant width and completeness are well known in fields like convexity, Banach space theory, and convex analysis. A compact, convex set K in ℝn (i.e., a convex body) is said to be of constant width if the distance between any two parallel supporting hyperplanes of K is the same. On the other hand, such a convex body K is called (diametrically) complete if any proper superset of it has larger diameter than K. It is obvious that these definitions can also be extended to any (normed or) Minkowski space, using the corresponding distance measures. In Euclidean spaces of any dimension as well as in arbitrary normed planes, constant width and completeness are equivalent. This is no longer the case in n-dimensional Minkowski spaces if n > 2, yielding the notion of perfect norms (used for norms in which this equivalence still holds). Surveys and basic references on bodies of constant width and complete bodies in Euclidean n-space are [8], [13], and [17], and results on their analogues in Minkowski spaces are collected or proved in [9], [22], [24], and [25].

A relatively new and closely related notion is that of reduced bodies: A convex body K in ℝn is said to be reduced if any convex proper subset of it has smaller (minimal) width. This notion creates already in Euclidean n-space sufficiently interesting open research problems; see the survey [19]. For example, in the non-planar case no reductions of the regular simplex are known and neither if such a reduction would be a minimizer of the volume for prescribed minimal width; cf. [16]. Moreover, only recently the question whether there exist reduced polytopes in Euclidean n-space if n ≥ 3 could be answered positively [10]. In [18] the notion of reducedness was extended to Minkowski spaces, see also [1] and the corresponding survey [20]. Since, in general, in such spaces complete bodies need not be reduced (see [23]) and one can construct reduced bodies which are not complete, the question is how these two classes are related to each other. Indeed, the family of bodies of constant width forms a subfamily of both. In this article we pose the question whether for non-perfect norms the family of bodies of constant width is precisely the intersection of the above two families. Moreover, it turned out in recent work (see, e.g., [5] and [7]) that it is advantageous to investigate the whole matter for generalized Minkowski spaces, in which the unit balls (called gauge bodies) are still convex, having the origin as interior point, but need not be 0-symmmetric.

To do so, we use several times the so called Minkowski asymmetry. There exists a rich variety of asymmetry measures for convex sets (see [14, section 6] and also [28] for a comprehensive overview), but amongst all, the one receiving most attention is the Minkowski asymmetry. In [5] it is shown how it naturally relates to complete and constant width sets in Minkowski spaces. Moreover, it has been repeatedly used to sharpen and strengthen geometric inequalities and related results; cf. [2], [7], [11] and [15].

We prove the following results for generalized Minkowski spaces: If the convex body K is a complete and reduced simplex, then it is of constant width. And the same implication holds for the large family of all convex bodies having a smooth extreme point (obviously, this class contains all strictly convex and all smooth convex bodies). Extending the notion of perfect norm to generalized Minkowski spaces, we also obtain some results on perfect gauge bodies, including a characterization of them via completions of the convex bodies under consideration.

2 Notation and background

By conv(A), int(A), bd(A), and aff(A) we denote the convex hull, the interior, the boundary, and the affine hull of a set A ⊂ ℝn, respectively, and we write [x, y] = conv({x, y}) for the line segment whose endpoints are x, y ∈ ℝn. We also use the notation [n] for {1, . . . , n}.

Let 𝒦n be the family of convex and compact sets (bodies) inℝn, and let K, C ∈ 𝒦n. We call K+C := {x+y : xK, yC} the Minkowski sum of K and C, and for any real ρ > 0 the set ρK := {ρx : xK} is the ρ-dilatation of K; we write −K := (−1)K. Any K ∈ 𝒦n such that K = cK for some c ∈ ℝn is called (centrally) symmetric, and 0-symmetric if c = 0.

If C ∈ 𝒦n with the origin 0 ∈ int(C), then C may be called a gauge body (or unit ball) of a generalized Minkowski space induced by C. Any non-negative function γ, which takes the value 0 only at the origin and satisfies γ(λx) = λγ(x) for all λ ≥ 0 and γ(x + y) ≤ γ(x)+ γ(y), is called a gauge function. Thus a gauge function meets all the requirements of a norm except for the symmetry γ(x) = γ(−x).

This means that the two definitions γ(x) := inf { λ ≥ 0 : xλC} for any given gauge body C and C := {x ∈ ℝn : γ(x) ≤ 1} for any given gauge function γ establish a one-to-one correspondence between gauge bodies and gauge functions similar to the well-known one-to-one correspondence between norms and 0-symmetric convex bodies with non-empty interior. Note that in the following we do not really assume 0 to be an interior point of C, as we do not use the gauge function; our considerations are, more generally, based on translation-invariant radius functions.

Denoting the Hausdorff distance by dH, we say that a sequence (Ai)i∈ℕ with Ai ⊂ ℝn converges to A ⊂ ℝn if limi→∞ dH(Ai , A) = 0.

The support function h(K, ⋅) : ℝn → ℝof a convex body K is defined by h(K, a) = max{aT x : xK}, where a ∈ ℝn. For b ∈ ℝ, we write Ha,b:={xn:aTxb} for the half-space with outer normal a and offset b, and Ha,b := {x ∈ ℝn : aT x = b} is written for the corresponding boundary hyperplane. The hyperplane Ha,b supports K at xK if xHa,b and KHa,b, which means that b = h(K, a). A point x ∈ bd(K) is extreme if x ∉ conv(K \ {x}), exposed if there exists a hyperplane supporting K at x only, and smooth if the hyperplane supporting K at x is unique. The set of all extreme points of K is denoted by ext(K).

The term Kopt C summarizes that KC and for all ρ < 1 and cC it holds that KρC + c. The circumradius R(K, C) of K with respect to C is the smallest λ ≥ 0 such that a translate of λC contains K. The inradius r(K, C) of K with respect to C is the largest λ ≥ 0 such that a translate of λC is contained in K. In case of a 0-symmetric C, the translations needed above are called the circumcenter and the incenter of K with respect to C, respectively.

The diameter D(K, C) of K with respect to C is defined as D(K, C) = 2 max{R([x, y], C) : x, yK}, the s-breadth bs(K, C), often also called s-width, of K with respect to C in the direction s ∈ ℝn \ {0} is bs(K, C) = 2h(KK, s)/h(CC, s). We will use several times that D(K, C) = maxs≠0 bs(K, C), which is shown to be true in [12] if C is 0-symmetric, but this obviously remains true for arbitrary convex bodies, since both, diameter and s-breadth, remain constant when replacing C by (1/2)(CC); see [7, Lemma 2.8]. The (minimal) width w(K, C) of K with respect to C is w(K, C) = mins≠0 bs(K, C).

The Minkowski asymmetry s(K) is the smallest λ ≥ 0 such that λK contains a translate of −K, i.e., s(K) = R(−K, K). Moreover, if for c ∈ ℝn the inclusion −(−c + K) ⊂ s(K)(−c + K) holds, then we say that c is the Minkowski center of K, and if c = 0 we say that K is Minkowski-centered.

One should notice that if c is a Minkowski center of K, we have −(−c + K) ⊂opt s(K)(−c + K). However, by the definition of s(K) the inclusion −(−c + K) ⊂ s(K)(−c + K) already implies the optimal inclusion. In such situations we will not use the “⊂opt”-symbol. It should be reserved to point out an assumption or conclusion about the inclusion that is not a general fact.

An n-simplex is the convex hull of n + 1 affinely independent points. It is well known that s(K) ∈ [1, n], and that s(K) = 1 holds if and only if K is symmetric, whereas s(K) = n holds if and only if K is an n-simplex; cf. [14].

A set K is complete with respect to C if D(Kʹ , C) > D(K, C) for every Kʹ ⊋ K, and K is reduced with respect to C if w(Kʹ , C) < w(K, C) for every Kʹ ⊊ K. With KK and KK we denote a completion or a reduction of K, respectively. This means in the first case that D(K, C) = D(K, C) as well as K is complete, and in the second that w(K, C) = w(K, C) as well as K is reduced. A set K is of constant width with respect to C if w(K, C) = D(K, C) or, equivalently, if KK = ρ(CC) (where ρ = D(K, C)/2 in this case). A set K is called pseudo-complete with respect to a symmetric C if D(K, C) = r(K, C) + R(K, C). One should observe that if C is symmetric, then any complete K is also pseudo-complete; see [24].

A gauge body C ∈ 𝒦n and the generalized Minkowski space induced by C are called perfect if K is of constant width with respect to C whenever K is complete with respect to C. By definition, in case that C = −C, the norm induced by C is called perfect iff C is perfect.

3 Completeness and reducedness

In Euclidean spaces of arbitrary dimension and in normed planes, completeness and constant width are equivalent notions (see [4], [9], and [22], as well as Lemma 4.6 below). Moreover, it is easy to see that any K of constant width with respect to an arbitrary body C is complete and reduced with respect to C. However, the contrary is, to the best of our knowledge, not known in general and has (also for norms) not been asked before. This question is the backbone of this article.

Open Question 3.1

. Let K, C ∈ 𝒦n be such that K is complete and reduced with respect to C. Does this imply that K is of constant width with respect to C?

The following lemma collects some facts about completeness and reducedness, showing that most of the problems may be reduced from arbitrary bodies to 0-symmetric ones.

Lemma 3.2

Let K, C ∈ 𝒦n. Then the following statements hold true.

  • (i) K is of constant width with respect to C iff K is of constant width with respect to CC.

  • (ii) K is complete with respect to C iff K is complete with respect to CC.

  • (iii) K is reduced with respect to C iff K is reduced with respect to CC.

  • (iv) C is perfect iff CC is perfect.

  • (v) There exist completions and reductions of K with respect to C.

  • (vi) If K is complete with respect to C, then every point x ∈ bd(K) is the endpoint of a diametrical segment.

  • (vii) If K is reduced with respect to C, then for every x ∈ ext(K) there exist yxK and s ∈ ℝn \ {0} such that bs([x, yx], C) = bs(K, C) = w(K, C) (see [20, Theorem 1] for the case that C = −C).

  • (viii) The set K is complete with respect to C iff K = ⋂x∈bd(K)(x + D(K, CC)(CC)) (spherical intersection property with respect to CC).

Proof. The first statement directly follows from the fact that K is of constant width with respect to C iff KK = (D(K, C)/2)(CC), and (ii) as well as (iii) directly follow from the fact that w(K, C) = 2w(K, CC) and D(K, C) = 2D(K, CC). Statement (iv) is a direct corollary of (i) and (ii), whereas the others follow from (ii) and (iii), taking into account that all those statements are well known for C = −C; see [9], [20], and [22].

Now we are able to state our first theorem confirming Open Question 3.1 in many particular cases, for instance, if the convex body is smooth or strictly convex.

Theorem 3.3

Let K, C ∈ 𝒦n be such that K is complete and reduced with respect to C and such that there exists a smooth extreme point x of K. Then K is of constant width.

Proof. By Lemma 3.2 (i), (ii) and (iii), we can assume that C is 0-symmetric. Since K is complete and x ∈ bd(K), we may use Lemma 3.2 (vi) to obtain that there exists yxK such that 2R([x, yx], C) = D(K, C). On the other hand, Lemma 3.2 (vii) implies that there exist two parallel supporting hyperplanes H±a,βi,i=1,2,a0, βi ∈ ℝ, at distance w(K, C) such that xHa,β1. Now, since [x, yx] is a diametrical segment, there exists s ≠ 0 such that bs(K, C) = D(K, C). Applying the smoothness of K at x, we obtain s = λa, λ > 0, and therefore w(K, C) = D(K, C).

The following proposition characterizes an optimal inclusion between two sets by their touching points; cf. Theorem 2.3 in [6].

Proposition 3.4

Let K, C ∈ 𝒦n. We have Kopt C iff KC and, for some 2 ≤ mn + 1, there exist p1, . . . , pmKC and hyperplanes Hai,1 supporting K and C at pi, i ∈ [m], such that 0 ∈ conv({a1, . . . , am}).

The following corollary combines optimal inclusion with the notion of Minkowski-centered polytopes.

Corollary 3.5

Let P ∈ 𝒦n be a Minkowski-centered polytope. Then

1+1sPconvPPPPsP+1PP,

and there exist vertices pi and facet normals ai of P, with i ∈ [m] for some 2 ≤ mn + 1, such that 0 ∈ conv({a1, . . . , am}) and ±(1 + 1/s(P))pi is a vertex of (1 + 1/s(P)) conv(P ∪ (−P)) contained in a facet of PP, which itself is completely contained in a facet of (s(P) + 1)(P ∩ (−P)), both with outer normalai.

Proof. Since 0 is the Minkowski center of P, we have −Popt s(P)P. Thus, by Proposition 3.4, there exist vertices pi of P and ai ≠ 0, i ∈ [m], satisfying 0 ∈ conv({a1, . . . , am}), such that Fi=Hai,1P is a facet of P with −pis(P)Fi for all i ∈ [m], for some m ∈ {2, . . . , n + 1}. Now, it obviously holds that ±Fi:=±(Fipi) is a facet of PP containing the vertex ±(1 + 1/s(P))pi of (1 + 1/s(P)) conv(P ∪ (−P)), which is also contained in the facet ±(s(P) + 1)(Fi ∩ (−P)) of (1 + s(P))(P ∩ (−P)). The inclusion of PP in (s(P) + 1)(P ∩ (−P)) proves that Fi is contained in (a facet of) (1 + s(P))(P ∩ (−P)).

Let us observe that the chain of inclusions (1+1s(K)) conv(K ∪ (−K)) ⊂ KK ⊂ (s(K) + 1)(K ∩ (−K)) in Corollary 3.5 remains true for non-polytopal K ∈ 𝒦n.

If K ⊂ ℝ3 is a regular tetrahedron with centroid at the origin, then Corollary 3.5 explains how the cube conv(K ∪ (−K)), the cuboctahedron KK, and the octahedron K ∩ (−K) can be placed such that the cube is optimally contained in the octahedron, and still the cuboctahedron fits in between; see Figure 1.

A cube optimally contained in an octahedron, and a cuboctahedron fitting in between.
Figure 1

A cube optimally contained in an octahedron, and a cuboctahedron fitting in between.

Next we state two propositions taken from [5, Lemma 2.5 and Corollary 2.10] which characterize pseudo-completeness (which becomes completeness if the container is an n-simplex).

Proposition 3.6

Let K, C ∈ 𝒦n be such that K is Minkowski-centered and C = −C. Then the following are equivalent:

  • (i) KKD(K, C)C ⊂ (s(K) + 1)(K ∩ (−K)).

  • (ii) K is pseudo-complete with respect to C.

Moreover, if K is complete with respect to C, then K satisfies both conditions above, and any of them implies that R(K, C)/D(K, C) = s(K)/(s(K) + 1); cf. [7, Corollary 6.3].

Proposition 3.7

Let S, C ∈ 𝒦n be such that S is a Minkowski-centered n-simplex and C = −C. Then the following are equivalent:

  • (i) S is pseudo-complete with respect to C.

  • (ii) SSD(S, C)C ⊂ (n + 1)(S ∩ (−S)).

  • (iii) S is complete with respect to C.

  • (iv) R(S, C)/D(S, C) = n/(n + 1) (equality case in Bohnenblust’s inequality, see [3] and [21]).

The proposition below is taken from [20, Corollary 7] and shows a quite similar structure for the reducedness of simplices as the one given in Proposition 3.7 for completeness.

Proposition 3.8

Let S, C ∈ 𝒦n be such that S is an n-simplex and C = −C. Then the following are equivalent:

  • (i) S is reduced with respect to C.

  • (ii) w(S, C)CSS touches all facets of SS with outer normals parallel to outer normals of facets of ±S.Putting Propositions 3.7 and 3.8 together, we obtain our second theorem, confirming Open Question 3.1 in the case of an n-simplex.

Theorem 3.9

Let S, C ∈ 𝒦n be such that S is a complete and reduced n-simplex with respect to C. Then S is of constant width with respect to C.

Proof. Without loss of generality, we can assume that C is 0-symmetric; see Lemma 3.2 (i), (ii), and (iii). The completeness of S implies by Proposition 3.7 that SSD(S, C)C ⊆ (n + 1)(S ∩ (−S)), and by Corollary 3.5 all facets of SS parallel to facets of S are contained in facets of (n + 1)(S ∩ (−S)). On the other hand, since S is reduced, Proposition 3.8 implies that w(S, C)CSS with touching points in all facets of SS which are parallel to facets of S. Hence w(S, C)C ⊂opt D(S, C)C and thus w(S, C) = D(S, C). 2

There is a natural connection between the equality case in the inequality of Leichtweiss (see [21]) and reduced sets, which is reflected in the following proposition.

Proposition 3.10

Let S, C ∈ 𝒦n be such that S is a Minkowski-centered n-simplex and C = −C. Then the following are equivalent:

  • (i) w(S, C)/r(S, C) = n + 1 (equality case in Leichtweiss’ inequality [21]).

  • (ii) (1+1/n) conv(S ∪ (−S)) ⊂ w(S, C)CSS, and (1+1/n)S touches SS in all facets with outer normals parallel to outer normals of facets of ±S in the points precisely given by Corollary 3.5.

It is immediate to observe that any pair of convex bodies S, C, where S is an n-simplex and C = −C, satisfying any condition in Proposition 3.10, fulfills also Proposition 3.8, thus implying that S is reduced with respect to C. However, the contrary is not true: there exist reduced simplices S with w(S, C)/r(S, C) < n + 1.

Further, let us ask the following: Assuming K, C ∈ 𝒦n to be such that K is Minkowski-centered, complete and reduced with respect to C, with C = −C, does this imply that (1+1s(K)) conv(K ∪ (−K)) ⊂ w(K, C)C? If the answer to this question would be affirmative, then, together with Proposition 3.6 and the trivial inclusion w(K, C)CKK, by Corollary 3.5 we would have that w(K, C)Copt D(K, C)C, and thus the answer to Open Question 3.1 would be affirmative, too.

Possibly the most important class of sets K not covered by Theorem 3.3 nor by Theorem 3.9 are polytopes. The aim of the remainder of this section is to give a positive answer to Open Question 3.1 in some cases when K, C ∈ 𝒦n are polytopes.

The two subsequent propositions are taken from [20, Corollary 1] and [25, Lemma 4], respectively.

Proposition 3.11

Let K, C ∈ 𝒦n be such that C = conv({±q1, . . . , ±qm}) is a 0-symmetric polytope and K is reduced with respect to C. Then K is a polytope, there exist ci ∈ ℝn such that

K=conv(im(ci+wK,C/2[qi,qi])),

and each segment ci + (w(K, C)/2)[−qi , qi] attains the width w(K, C) of K with respect to C.

Proposition 3.12

Let K, C ∈ 𝒦n be such that C=j[l]H±aj,1,ajn,j[l], is a 0-symmetric polytope and K is complete with respect to C. Then K is a polytope, there exist dj ∈ ℝn, j ∈ [l], such that

K=j[l](dj+H±aj,1),

and the diameter D(K, C) is attained in every direction aj, j ∈ [l].

Let K, C ∈ 𝒦n. It follows directly from the definition of the width that some translate of w(K, C)C is optimally contained in KK. Now assume that C := conv({±q1, . . . , ±qm}) is a 0-symmetric polytope and that K is reduced with respect to C. Then Proposition 3.11 implies that all vertices ±w(K, C)qi of w(K, C)C belong to bd(KK). Thus Proposition 3.11 strengthens the optimal inclusion of w(K, C)C in KK, which only assures a certain distribution of the vertices of w(K, C)C touching the boundary of KK.

Analogously, Proposition 3.12 implies in case of a 0-symmetric polytope C, that if K is complete then K must be a polytope and all those facets of KK that are parallel to facets of K are contained in facets of D(K, C)C.

Generalizing the above to (polytopal) gauge bodies C, which are possibly not 0-symmetric, one may “just” replace the vertices/facets of C by those of CC in the representation of a reduced/complete set K, respectively. However, while the vertices of CC are simply differences of vertices of C, the facet structure of CC relies not only on that of C.

The following two lemmas solve Open Question 3.1 for polytopes K, C ∈ 𝒦n for certain configurations of vertices/facets between K and C.

Lemma 3.13

Let K, C ∈ 𝒦n be such that C is a 0-symmetric polytope and K is complete and reduced with respect to C (from which we know that K is a polytope by Propositions 3.11 and 3.12). Then the existence of a vertex of w(K, C)C belonging to a facet of KK parallel to a facet of K implies that K is of constant width with respect to C.

Proof. This follows directly from Proposition 3.12 as the completeness implies that facets of KK parallel to those of K have to be contained in facets of D(K, C)C. Hence w(K, C)Copt D(K, C)C and thus w(K, C) = D(K, C).

Lemma 3.14

Let K, C ∈ 𝒦n be such that C is a 0-symmetric polytope and K is complete and reduced with respect to C. Using the notation in Propositions 3.11 and 3.12, let i ∈ [m] be such that ci + (w(K, C)/2)qi is a vertex of K and ci − (w(K, C)/2)qi belongs to the relative interior of a facet Fj=(dj+Haj,1)KofK,j[l]. Then K is of constant width.

Proof. On the one hand, Proposition 3.11 implies that the segment ci + (w(K, C)/2)[−qi , qi] attains the width w(K, C) in some direction. Since ci − (w(K, C)/2)qi belongs to the relative interior of Fj, we have w(K, C) = w(ci+(w(K,C)/2)[qi,qi],C)=baj(K,C). On the other hand, since K is complete, every point in bd(K) is an endpoint of a diametrical segment (cf. Lemma 3.2 (vi)), and thus also ci − (w(K, C)/2)qi is such an endpoint. However, since the only hyperplane supporting K at ci − (w(K, C)/2)qi is dj+Haj,1, we obtain D(K,C)=baj(K,C) and therefore w(K, C) = D(K, C).

4 Perfect gauge bodies

In the following, we connect Open Question 3.1 with perfect gauge bodies (and thus with perfect norms in case that these bodies are symmetric). Let us first state a quite obvious observation: If Open Question 3.1 holds true, then a gauge body is perfect if and only if completeness implies reducedness.

The final lemma in [9] gives in its negation a necessary condition for a 3-dimensional polytopal norm to be perfect:

Proposition 4.1

Any perfect 0-symmetric polytopal body C ∈ 𝒦3 is simple (i.e., every vertex of C is contained in at most three facets).

The following lemma extends Proposition 4.1 to higher-dimensional spaces following in its proof the idea of Eggleston’s original proof of Proposition 4.1

Lemma 4.2

Let n ≥ 3 and C ∈ 𝒦n be a 0-symmetric polytope. If C is perfect, then every pair of non-disjoint facets F1, F2 of C intersects in at least an edge of C.

Proof. Let us assume that F1, F2 are two facets of C only intersecting in a vertex v of C. The intersection aff(F1)∩ aff(F2) is an affine (n − 2)-subspace. Let a ∈ ℝn, and let Ha,1 be a hyperplane supporting C solely at v and containing aff(F1)∩ aff(F2), such that CHa,1. Let ε > 0 be small enough such that Ha,1−ε intersects F1 and F2 in (n − 2)-dimensional polytopes and xi ∈ relint(FiHa,1−ε), i = 1, 2. Then Ha,1−ε is a supporting hyperplane of the intersection C ∩ (x2x1 + C) and X := C ∩ (x2x1 + C) ∩ Ha,1−ε is an (n − 2)-dimensional polytope.

We now define the set Y := conv({0, x2x1}∪ X). If ε tends to 0, then [0, x2x1] and X converge to 0 and v, respectively. On the other hand, for any xXF1 ⊂ bd(C) we have that [−x, x] ⊂opt C, hence D([0, x], C) = 1. Analogously, x1x2 + xx1x2 + XF2 ⊂ bd(C), thus [x1x2 + x, −(x1x2 + x)] ⊂opt C, and hence D([x2x1, x], C) = D([x1x2 + x, 0], C) = 1. Since the diameter D(Y, C) is always attained by a pair of extreme points, we conclude that D(Y, C) = 1.

Now let Y be a completion of Y with respect to C. Using the spherical intersection property (Lemma 3.2 (viii)), we obtain Y*=xY*(x+C)C(x2x1+C), and since Ha,1−ε supports C ∩ (x2x1 + C) at X, it also supports Y at X. Hence the supporting hyperplanes of YY parallel to Ha,1−ε support it at a set of dimension at least n − 2, whereas C is only supported at ±v. This proves YYC, and hence Y is complete but not of constant width with respect to C.

Lemma 4.2 in fact extends Proposition 4.1 to arbitrary dimensions n ≥ 3. Indeed, it is easy to see that the property that every pair of non-disjoint facets F1, F2 of C intersects in at least an edge of C in 3-space is equivalent to C being simple.

Let us point out that this equivalence gets lost in higher dimensions: It still holds that every two non-disjoint facets of a simple polytope intersect in an (n−2)-dimensional face of P. But for the converse consider a double pyramid P ⊂ ℝn with an (n − 1)-simplex as its base. While every two different facets of P must have at least n − 2 vertices in common, P is not simple if n ≥ 4.

Corollary 4.3

Let n ≥ 3 and S, C ∈ 𝒦n be such that S is an n-simplex and CC = SS. Then C is not perfect.

Proof. Because of Lemma 3.2 (iv), it is enough to verify the corollary in case C = SS, and we may assume, without loss of generality, that S is Minkowski-centered.

Let S = conv({p1, . . . , pn+1}) and let Fi = conv({pj : j ∈ [n + 1] \ {i}}) be the facet of S not containing pi. Now consider the two facets −p1 + F1 and p2F2 of SS, which have the same outer normals as F1 and −F2, respectively, and p2p1 as a common vertex. From Corollary 3.5 we obtain −p1 + F1 ⊂ (n + 1)(F1 ∩ (−S)) and p2F2 ⊂ (n + 1)(S ∩ (−F2)). Indeed, since S ∩ bd(−nS) = {p1, . . . , pn+1}, we have F1 ∩ (bd(−nS))) = {p2, . . . , pn+1}. Thus

(p1+F1)(relbd(n+1)(F1)(S))))=p1+{p2,....,pn+1}

and, analogously, (p2F2)∩ (relbd((n+1)(S∩ (−F2)))) = p2−{p1, p3, . . . , pn+1}. Hence (−p1+F1)∩ (p2F2) = {p2p1}, which finishes the proof because of Lemma 4.2.

Corollary 4.4

Let n ≥ 3 and S, C ∈ 𝒦n be such that C is 0-symmetric and S is an n-simplex with SSD(S, C)C ⊂ (n + 1)(S ∩ (−S)). Then C is not perfect.

Proof. By Proposition 3.7 we know that if D(S, C)CSS, then S is complete but not of constant width with respect to C, showing the non-perfectness of C. However, in case that D(S, C)C = SS, Corollary 4.3 implies that C is non-perfect either.

Finally, the following theorem gives a characterization of perfect norms in terms of the linearity of the width for K and any completion K.

Theorem 4.5

Let C ∈ 𝒦n. Then the following are equivalent:

  • (i) C is perfect.

  • (ii) For all K ∈ 𝒦n and any completion K of K we have w(K + K, C) = w(K, C) + w(K, C).

Proof. By Lemma 3.2 (iv) we may assume, without loss of generality, that C is 0-symmetric. We observe that

w(K+K*,C)=minsbs(K+K*,C)=mins(K,C)+bs(K*,C))minsbs(k,C)+minsbs(k*,C)=w(k,C)+w(K*,C).

Now if (i) holds, the set K is of constant width, and hence there is equality above because bs(K, C) = w(K, C) for all s, therefore proving (ii).

On the contrary let us assume that (i) is false. Then we will prove that (ii) is false as well. Assuming that (i) is false, there exists a complete body U which is not of constant width. The idea of the proof is to construct a subset K of U such that U is a completion of K and the widths w(K, C) and w(U, C) are not achieved in the same direction. This then implies w(K + U, C) > w(K, C) + w(U, C), thus leading to the desired contradiction.

For 0-symmetric C any complete set has coinciding in- and circumcenters (see, e.g., [26]). Hence we may assume again, without loss of generality, that r(U, C)CUR(U, C)C. Now by Proposition 3.4 there exist points piU ∩ bd(R(U, C)C) and outer normals ai of hyperplanes supporting U and R(U, C)C at pi, i ∈ [m], 2 ≤ mn+1, with 0 ∈ conv({a1, . . . , am}). Moreover, we may scale the vectors ai such that (ai)Tpi = R(U, C). Defining qi := −(r(U, C)/R(U, C))pir(U, C)CU, we see that all the segments [pi , qi] are diametrical chords of U. Hence it must hold that bai(U,C)=D(U,C) for all i ∈ [m] and, defining β2 := h(r(U, C)C, a2) = (r(U, C)/R(U, C))(a2)Tp2, also that P1Ha2,β2.

Since 0 = i=1mλiai,λi>0, we may assume that (a2)Tp1 < 0. Now consider the setM=UH±a2,β2, which still contains p1 and q1. The half-space Ha2,β2 supports the inball r(U, C)C and therefore contains extreme points of it. By continuity of ba(U, C), a ∈ ℝn \ {0}, we have ba2(U,C)ba(U,C)<D(U,C)w(U,C) for any a sufficiently close to a2. However, since the set of exposed points is dense in the set of extreme points of any convex body, we may carefully choose a such that the following are true:

  • w(U, C) < ba(U, C),

  • H±a2,β2 are hyperplanes supporting r(U, C)C solely at a pair of exposed antipodal points of the inball r(U, C)C of U, and

  • p1,q1H±a2,β2<.

Due to these conditions, there exists ε > 0 small enough such that the set K:=UH±a2,β2εa2 still satisfies p1, q1K, where ‖a2 denotes the Euclidean norm of a. Now, because D(K, C) = D([p1, q1], C) = D(U, C), we see that U is a completion of K. Moreover, since each of the half-spaces H±a2,β2εa2 touches (r(U, C) − ε)C = r(K, C)C in a unique exposed point and r(K, C)C ⊂ int(U), we have ba(K, C) = 2r(K, C) < bs(K, C) for all directions sa. Thus the width of K is uniquely attained in the direction a, and because w(U, C) < ba(U, C), we can conclude that w(K + U, C) > w(K, C) + w(U, C).

Theorem 4.5 generalizes [5, Lemma 2.2], which only considered the Euclidean case.

The following lemma shows that all two-dimensional generalized Minkowski spaces are perfect, cf. [9, p. 171] for normed spaces.

Lemma 4.6

Let K, C ∈ 𝒦2. If K is complete with respect to C, then K is of constant width with respect to C.

Proof. By Lemma 3.2 we can assume, without loss of generality, that C is 0-symmetric. First we consider the case where K and C are polygons. Let xK be a point in the relative interior of the edge E of K induced by Ha,1. Then ba(K, C) = D(K, C) (which follows from the uniqueness of the outer normal, up to positive multiples, at that point and the completeness of K) means that the edges of KK parallel to edges of K are contained in the boundary of D(K, C)C. Since for n = 2 all edges of KK are parallel to edges of K it follows that KK = D(K, C)C.

Now, let K and C be arbitrary, planar convex sets, let a be unit vector such that ba(K, C) = D(K, C), and tj , sj ∈ ℝ, j ∈ [2], such that Ha,tj,Ha,sj, j ∈ [2], are the parallel supporting lines of K and C with outer normal a. We then consider a sequence Ci of polygons with CCijHa,sj and CiC (i → ∞). Then

D(K,Ci)D(K,C)=ba(K,C)=ba(K,Ci)D(K,Ci),

and therefore D(K, Ci) = D(K, C) for all i.

Now we assume KiK to be a completion of K with respect to Ci. We prove by contradiction that KiK (i → ∞). If not, let us observe that since Ki is a completion of K, we have Kix + (D(K, C)/2)(CC) for some xK. Hence {Ki}i∈ℕ is bounded. By the Blaschke Selection Theorem [27, Theorem 1.8.7], there exists a subsequence of Ki (we may assume that it is the sequence itself) such that KiK0K (i → ∞). Then on the one hand the completeness of K with respect to C tells us that D(K, C) < D(K0, C), and on the other hand the continuity of the diameter implies that limi→∞ D(Ki , Ci) = D(K0, C). Altogether this shows that

D(K,C)<D(K0,C)=limiD(Ki,Ci)=D(K,C),

which is a contradiction. Thus KiK (i → ∞), and hence KK = limi→∞ Ki−limi→∞ Ki = limi→∞(KiKi) = limi→∞(D(Ki , Ci)Ci) = D(K, C)C.

Let us observe that the argument for general K and C in Lemma 4.6 uses the fact that Open Question 3.1 is true for polygons. Indeed, it holds true for every 0-symmetric C ∈ 𝒦n, with n ∈ ℕ. This therefore motivates why examples of non-perfect norms are polytopal. Moreover, this approach might be useful as well when considering sets that are complete and reduced simultaneously.

Acknowledgements

We would like to thank Matthias Henze for helping to correct a mistake in the example after Lemma 4.2. We would also like to thank the anonymous referee for her/his helpful comments improving the paper.

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About the article


Received: 2016-02-27

Revised: 2016-09-09

Published Online: 2019-01-18

Published in Print: 2019-01-28


Funding The second author was partially supported by Fundación Séneca, Science and Technology Agency of the Región de Murcia, through the Programa de Formación Postdoctoral de Personal Investigador, project reference 19769/PD/15, and the Programme in Support of Excellence Groups of the Región de Murcia, Spain, project reference 19901/GERM/15, and MINECO project reference MTM2015-63699-P, Spain.


Citation Information: Advances in Geometry, Volume 19, Issue 1, Pages 31–40, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0058.

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