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* ∈ 𝒦^{n} 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* ∈ 𝒦^{n}. 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 C* − *C*.

(ii) *K is complete with respect to C iff K is complete with respect to C* − *C*.

(iii) *K is reduced with respect to C iff K is reduced with respect to C* − *C*.

(iv) *C is perfect iff C* − *C 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 y*_{x} ∈ *K and s* ∈ ℝ^{n} \ {0} *such that b*_{s}([*x*, *y*_{x}], *C*) = *b*_{s}(*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*, *C* − *C*)(*C* − *C*)) (*spherical intersection property with respect to C* − *C*).

*Proof*. The first statement directly follows from the fact that *K* is of constant width with respect to *C* iff *K*−*K* = (*D*(*K*, *C*)/2)(*C* − *C*), and (ii) as well as (iii) directly follow from the fact that *w*(*K*, *C*) = 2*w*(*K*, *C* − *C*) and *D*(*K*, *C*) = 2*D*(*K*, *C* − *C*). 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* ∈ 𝒦^{n} 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 *y*_{x} ∈ *K* such that 2*R*([*x*, *y*_{x}], *C*) = *D*(*K*, *C*). On the other hand, Lemma 3.2 (vii) implies that there exist two parallel supporting hyperplanes ${H}_{\pm a,{\beta}_{i},\text{\hspace{0.17em}}}i=1,2,a\ne 0,$ *β*_{i} ∈ ℝ, at distance *w*(*K*, *C*) such that $x\in {H}_{a,{\beta}_{1}}.$ Now, since [*x*, *y*_{x}] is a diametrical segment, there exists *s* ≠ 0 such that *b*_{s}(*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* ∈ 𝒦^{n}. We have K ⊂^{opt} *C iff K* ⊂ *C and, for some* 2 ≤ *m* ≤ *n* + 1*, there exist p*^{1}, . . . , *p*^{m} ∈ *K* ∩ *C and hyperplanes* ${H}_{{a}^{i},1}$ *supporting K and C at p*^{i}, i ∈ [*m*]*, such that* 0 ∈ conv({*a*^{1}, . . . , *a*^{m}}).

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

#### Corollary 3.5

*Let P* ∈ 𝒦^{n} be a Minkowski-centered polytope. Then

$$\left(1+\frac{1}{s\left(P\right)}\right)\text{conv}\left(P\cup \left(-P\right)\right)\subset P-P\subset \left(s\left(P\right)+1\right)\left(P\cap \left(-P\right)\right),$$

*and there exist vertices p*^{i} and facet normals a^{i} of P, with i ∈ [*m*] *for some* 2 ≤ *m* ≤ *n* + 1*, such that* 0 ∈ conv({*a*^{1}, . . . , *a*^{m}}) *and* ±(1 + 1/*s*(*P*))*p*^{i} is a vertex of (1 + 1/*s*(*P*)) conv(*P* ∪ (−*P*)) *contained in a facet of P* − *P, which itself is completely contained in a facet of* (*s*(*P*) + 1)(*P* ∩ (−*P*))*, both with outer normal* ∓*a*^{i}.

*Proof*. Since 0 is the Minkowski center of *P*, we have −*P* ⊂^{opt} *s*(*P*)*P*. Thus, by Proposition 3.4, there exist vertices *p*^{i} of *P* and *a*^{i} ≠ 0, *i* ∈ [*m*], satisfying 0 ∈ conv({*a*^{1}, . . . , *a*^{m}}), such that ${F}_{i}={H}_{{a}^{i},1}\cap P$ is a facet of *P* with −*p*^{i} ∈ *s*(*P*)*F*_{i} for all *i* ∈ [*m*], for some *m* ∈ {2, . . . , *n* + 1}. Now, it obviously holds that $\pm {F}_{i}{}^{\prime}:=\pm \left({F}_{i}-{p}^{i}\right)$ is a facet of *P* − *P* containing the vertex ±(1 + 1/*s*(*P*))*p*^{i} of (1 + 1/*s*(*P*)) conv(*P* ∪ (−*P*)), which is also contained in the facet ±(*s*(*P*) + 1)(*F*_{i} ∩ (−*P*)) of (1 + *s*(*P*))(*P* ∩ (−*P*)). The inclusion of *P* − *P* in (*s*(*P*) + 1)(*P* ∩ (−*P*)) proves that ${{F}^{\prime}}_{i}$ is contained in (a facet of) (1 + *s*(*P*))(*P* ∩ (−*P*)).

Let us observe that the chain of inclusions $(1+\frac{1}{s\left(K\right)})$ conv(*K* ∪ (−*K*)) ⊂ *K* − *K* ⊂ (*s*(*K*) + 1)(*K* ∩ (−*K*)) in Corollary 3.5 remains true for non-polytopal *K* ∈ 𝒦^{n}.

If *K* ⊂ ℝ^{3} is a regular tetrahedron with centroid at the origin, then Corollary 3.5 explains how the cube conv(*K* ∪ (−*K*)), the cuboctahedron *K* − *K*, 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.

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* ∈ 𝒦^{n} be such that K is Minkowski-centered and C = −*C. Then the following are equivalent*:

*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* ∈ 𝒦*n 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) *S* − *S* ⊂ *D*(*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* ∈ 𝒦*n 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*)*C* ⊂ *S* − *S touches all facets of S* − *S 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* ∈ 𝒦*n 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 *S* − *S* ⊆ *D*(*S*, *C*)*C* ⊆ (*n* + 1)(*S* ∩ (−*S*)), and by Corollary 3.5 all facets of *S* − *S* 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*)*C* ⊆ *S* − *S* with touching points in all facets of *S* − *S* 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* ∈ 𝒦*n 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*)*C* ⊂ *S* − *S, and* (1+1/*n*)*S touches S* − *S 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* ∈ 𝒦^{n} to be such that *K* is Minkowski-centered, complete and reduced with respect to *C*, with *C* = −*C*, does this imply that $(1+\frac{1}{s\left(K\right)})$ 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*)*C* ⊂ *K* − *K*, by Corollary 3.5 we would have that *w*(*K*, *C*)*C* ⊂^{opt} *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* ∈ 𝒦^{n} are polytopes.

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

#### Proposition 3.11

*Let K*, *C* ∈ 𝒦^{n} be such that C = conv({±*q*^{1}, . . . , ±*q*^{m}}) *is a* 0*-symmetric polytope and K is reduced with respect to C. Then K is a polytope, there exist c*^{i} ∈ ℝ^{n} such that

$$$$$$K=\text{conv(}{\cup}_{i\in \left[m\right]}({c}^{i}+\left(w\left(K,C\right)/2\right)[-{q}^{i},{q}^{i}])),$$

*and each segment c*^{i} + (*w*(*K*, *C*)/2)[−*q*^{i} , *q*^{i}] *attains the width w*(*K*, *C*) *of K with respect to C*.

#### Proposition 3.12

*Let K*, *C* ∈ 𝒦^{n} be such that $C=\cap j\in \left[l\right]{H}_{\pm {a}^{j},1}^{\le},{a}^{j}\in {\mathbb{R}}^{n},j\in \left[l\right],$ *is a* 0*-symmetric polytope and K is complete with respect to C. Then K is a polytope, there exist d*^{j} ∈ ℝ^{n}, j ∈ [*l*]*, such that*

$$K=\underset{j\in \left[l\right]}{\cap}\left({d}^{j}+{H}_{\pm {a}^{j},1}^{\le}\right),$$

*and the diameter D*(*K*, *C*) *is attained in every direction a*^{j}, j ∈ [*l*].

Let *K*, *C* ∈ 𝒦^{n}. It follows directly from the definition of the width that some translate of *w*(*K*, *C*)*C* is optimally contained in *K* − *K*. Now assume that *C* := conv({±*q*^{1}, . . . , ±*q*^{m}}) 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*)*q*^{i} of *w*(*K*, *C*)*C* belong to bd(*K* − *K*). Thus Proposition 3.11 strengthens the optimal inclusion of *w*(*K*, *C*)*C* in *K* − *K*, which only assures a certain distribution of the vertices of *w*(*K*, *C*)*C* touching the boundary of *K* − *K*.

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 *K* − *K* 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 *C*−*C* in the representation of a reduced/complete set *K*, respectively. However, while the vertices of *C* − *C* are simply differences of vertices of *C*, the facet structure of *C* − *C* relies not only on that of *C*.

The following two lemmas solve Open Question 3.1 for polytopes *K*, *C* ∈ 𝒦^{n} for certain configurations of vertices/facets between *K* and *C*.

#### Lemma 3.13

*Let K*, *C* ∈ 𝒦^{n} 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 K* − *K 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 *K* − *K* parallel to those of *K* have to be contained in facets of *D*(*K*, *C*)*C*. Hence *w*(*K*, *C*)*C* ⊂^{opt} *D*(*K*, *C*)*C* and thus *w*(*K*, *C*) = *D*(*K*, *C*).

#### Lemma 3.14

*Let K*, *C* ∈ 𝒦^{n} 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 c*^{i} + (*w*(*K*, *C*)/2)*q*^{i} is a vertex of K and c^{i} − (*w*(*K*, *C*)/2)*q*^{i} belongs to the relative interior of a facet ${F}^{j}=\left({d}^{j}+{H}_{{a}^{j},1}\right)\cap K\text{\hspace{0.17em}}\text{\hspace{0.17em}}of\text{\hspace{0.17em}}K,j\in \left[l\right].$ *Then K is of constant width*.

*Proof*. On the one hand, Proposition 3.11 implies that the segment *c*^{i} + (*w*(*K*, *C*)/2)[−*q*^{i} , *q*^{i}] attains the width *w*(*K*, *C*) in some direction. Since *c*^{i} − (*w*(*K*, *C*)/2)*q*^{i} belongs to the relative interior of *F*_{j}, we have *w*(*K*, *C*) = $w({c}^{i}+(w(K,C)/2)\left[-{q}^{i},{q}^{i}\right],C)={b}_{{a}^{j}}\left(K,C\right).$ 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 *c*^{i} − (*w*(*K*, *C*)/2)*q*^{i} is such an endpoint. However, since the only hyperplane supporting *K* at *c*^{i} − (*w*(*K*, *C*)/2)*q*^{i} is ${d}^{j}+{H}_{{a}^{j},1},$ we obtain $D\left(K,C\right)={b}_{{a}^{j}}\left(K,C\right)$ and therefore *w*(*K*, *C*) = *D*(*K*, *C*).

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