ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.08 / 637–648
https://doi.org/10.26493/1855-3974.2559.e4f
(Also available at http://amc-journal.eu)
Configured polytopes and extremal
configurations
Tibor Bisztriczky
Department of Mathematics and Statistics, University of Calgary, Canada
Gyivan Lopez-Campos , Deborah Oliveros *
Instituto de Matemáticas, Universidad Nacional Autónoma de México,
Boulevard Juriquilla 3001, Juriquilla, Querétaro, 076230
Received 17 February 2021, accepted 7 January 2022, published online 19 August 2022
Abstract
We examine a class of involutory self-dual convex polytopes with a specified sets of
diameters, compare their vertex sets to extremal Lenz configurations, and present some of
their realizations.
Keywords: Involutory self-dual polytopes, configured polytopes, Lenz configurations, extremal con-
figurations.
Math. Subj. Class. (2020): 52-01, 52A15, 52B11
1 Introduction
We describe points in Rd by standard coordinates (x1, x2, . . . , xd). For 3 ≤ i ≤ d,
let Hi(bi) denote the hyperplane xi = bi, and Le(be+1, . . . , bd) = ∩di=e+1Hi(bi), e =
2, . . . , d − 1. Le(be+1, . . . , bd) is an e-flat, and denote the (e − 1)-sphere with centre c
and radius t in Le(be+1, . . . , bd) by Se−1(c, t). We denote the origin of Rd by cd, and let
(λw, p) := λw + (0, . . . , 0, p), for a point w ∈ Hd(0) = Ld−1(0) and {λ, p} ⊂ R.
Let Y be a set of points in Rd. Then conv(Y ) and aff(Y ) denote, respectively, the con-
vex hull and the affine hull of Y . For sets Y1, Y2, . . . Yn, let [Y1, Y2, . . . Yn] = conv (∪ni=1Yi)
and ⟨Y1, Y2, . . . Yn⟩ = aff (∪ni=1Yi). If Y = {y1, y2, . . . , yn} is finite, we let
[y1, y2, . . . , yn] = conv(Y ) and ⟨y1, y2, . . . , yn⟩ = aff(Y ).
*Corresponding author. Supported by Proyecto PAPIIT IG100721, 106318 and CONACYT Ciencia Básica
282280.
E-mail addresses: tbisztri@ucalgary.ca (Tibor Bisztriczky), gyivan.lopez@im.unam.mx (Gyivan
Lopez-Campos), doliveros@im.unam.mx (Deborah Oliveros)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
638 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648
Let P ⊂ Rd denote a convex d−polytope with L(P ) and Fi(P ), 0 ≤ i ≤ d−1, denot-
ing the face lattice and the set of i−faces of P . We let fi(P ) = |Fi(P )|, V (P ) = F0(P )
and F(P) = Fd−1(P ), assume familiarity with the basic notions of convex polytopes, and
refer to [3, 6] and [18] for basic terminology and definitions. Specifically, two polytopes
P1 and P2 are combinatorially equivalent (P1 ∼= P2) if there is an isomorphism (inclusion
preserving) from L(P1) to L(P2), and are dual if there is an anti-isomorphism (inclusion
reversing) from L(P1) to L(P2). If there is an anti-isomorphism Φ from L(P ) to L(P )
then P is self-dual, moreover, if Φ2 = id then P is involutory self-dual.
Let P ⊂ Rd be involutory self-dual via the anti-isomorphism on L(P ) induced by the
map v → v∗ with v ∈ V (P ), v∗ ∈ F(P ) and v /∈ v∗. A segment [v, w], with end-points
v and w, both vertices of P and with w ∈ v∗, is called a principal diagonal of P and
let D(P ) denote the set of principal diagonals of P . Finally, we say that P is configured
if each principal diagonal in P has length diam(P ), and that P is strictly configured if it
is configured and only principal diagonals of P have length diam(P ). We note that odd
regular polygons are strictly configured.
Let Xn ⊂ Rd be a set of n > d ≥ 2 points and Md(Xn) be the number of pairs
{x, y} ⊂ Xn such that diam(Xn) = ∥x− y∥, the distance between x and y. Let M(d, n)
be the maximum of Md(Xn) over all Xn ⊂ Rd. Then Xn is an extremal configuration if
Md(Xn) = M(d, n).
The problem of determining M(d, n) is due to Erdős in [4]. We list contributions to
the problem in the References, with specific mention of [11, 12] and [17] and the following
results:
(1) M(2, n) = n, and Xn ⊂ R2 is extremal if and only if V (P ) ⊆ Xn ⊆ bd(P ) for
some Reuleaux polygon P .
(2) M(3, n) = 2n − 2 and Xn ⊂ R3 is extremal if and only if Xn is the vertex set of
certain types of polytopal (Reuleaux) ball polytopes.
(3) M(d, n), d ≥ 4, grows quadratically with n, and extremal Xn are attained only by
Lenz Constructions.
In this last regard, we note (cf. [17]) that an (even dimensional) Lenz Configuration in
Rd, d = 2p ≥ 2, is any translate of a finite subset of ∪pi=1Ci where Ci is a circle with
centre at the origin O and radius ri, so that r2j + r
2
k = 1 for all j, k and the subspaces Ui,
spanned by Ci, yield the orthogonal decomposition Rd = U1 ⊕ U2 ⊕ ... ⊕ Up. For odd
dimensions d = 2p+ 1, C1 is replaced by a 2-sphere with centre O and radius r = 1√2 .
Theorem 1.1 (K. Swanepoel). For each d ≥ 4, there exists a number N(d) such that all
extremal configurations Xn, with n ≥ N(d), are Lenz Configurations.
We note that in [17], Swanepoel also determines M(d, n) for sufficiently large n.
Our interests in this paper are realizations (constructions) of strictly configured d-
polytopes P , d ≥ 3, and values of Md(P ) (number of principal diagonals of P ). In
Section 2, we will show that for strictly configured 4-polytopes there is a formula similar
to 1) and 2) that depends on the number of vertices and edges; furthermore we show the
convex hull of vertices of an extremal Lenz configuration is never a configured d-polytope.
The former raises the question of whether in dimension d ≥ 4 the situation for M(d, n)
may have at least another possible scenario, if the points are not in Lenz configurations. In
T. Bisztriczky et al.: Configured polytopes and extremal configurations 639
Section 3 we will give constructions of configured d-polytopes P for d ≥ 3 such that for
d = 4, M4(P ) ≤ 4n. These constructions consist of two steps: determining self-dual poly-
topes so that all principal diagonals have length (say 1), and then showing that the diameter
of the polytope is 1.
2 Principal diagonals
In this section, we assume that P ⊂ Rd is an involutory self-dual d-polytope via the anti-
isomorphism on L(P ) induced by v ∈ V (P )→ v∗ ∈ F(P ), and recall that D(P ) denotes
the set of principal diagonals of P .
Theorem 2.1. Let P ⊂ R3 be a configured 3-polytope. Then P is strictly configured and
extremal, that is, |D(P )| = 2f0(P )− 2.
Proof. Since P is self-dual, we have that f0(P ) = f2(P ) and so, f1(P ) = 2f0(P )− 2 by
Euler’s Theorem.
Let v ∈ V (P ). Then v∗ ∈ F2(P ) is a polygon and f0(v∗) = f1(v∗). On the one hand,
f0(v
∗) = |{g ∈ D(P ) | v ∈ g}| by definition. On the other hand, v ∈ e ∈ F1(P ) iff
e∗ ∈ F1(v∗), and so, f1(v∗) = |{e ∈ F1(P ) | v ∈ e}|. Thus |{g ∈ D(P ) | v ∈ g}| =
|{e ∈ F1(P ) | v ∈ e}| and |D(P )| = |F1(P )|.
Theorem 2.2. Let P ⊂ R4 be a strictly configured 4-polytope. Then |D(P )| ≤ 2f1(P )−
2f0(P ).
Proof. Let V (P ) = {v1, ..., vn} and F1(P ) = {e1, ..., em}. Then F2(P ) = {e∗1, ..., e∗m}
and F(P ) = {v∗1 , ..., v∗n} by the self-duality of P .
We recall from [1] that fjk(P ), 0 ≤ j < k ≤ 3, is the number of pairs of j-faces
Gj and k-faces Gk such that Gj ⊂ Gk, and that f02(P ) ≤ 6f1(P ) − 6f0(P ). By the
self-duality of P , we have also that
n∑
i=1
f1(v
∗
i ) = f13(P ) = f02(P ),
n∑
i=1
f2(v
∗
i ) = f23(P ) = f01(P ) and
f01(P ) =
m∑
j=1
f0(ej) = 2f1(P )
Finally, let v ∈ V (P ) and e ∈ D(P ) of a configured P ⊂ R4.
Then v ∈ e if, and only if, e = [v, w] and w ∈ F0(v∗). Thus, f0(v∗) is the number
of principal diagonals of P that contain v, and
∑n
i=1 f0(v
∗
i ) = 2|D(P )|. Then by Euler’s
Theorem,
|D(P )| = 1
2
n∑
i=1
(2 + f1(v
∗
i )− f2(v∗i ))
= n+
1
2
n∑
i=1
f1(v
∗
i )−
1
2
n∑
i=1
f2(v
∗
i )
= f0(P ) +
1
2
f02(P )−
1
2
f01(P )
≤ f0(P ) + [3f1(P )− 3f0(P )]− f1(P ).
(2.1)
640 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648
End of Theorem 2.2.
We let Md(Q) = Md(V (Q)) for a d−polytope Q, and observe that if P ⊂ R4 is strictly
configured then M4(P ) is linear in f1(P ) and f0(P ). This raises the following question: Is
there a set of n vertices of a strictly configured polytope in Lenz Configuration? We show
below that the answer is no if f0(P ) > 5; in fact, we present in Section 3 a subfamily of
such P ⊂ R4 with f1(P ) ≤ 3f0(P ) and M4(P ) ≤ 4f0(P ).
If n = 5 and d = 4, it is easy to prove that the polytope with vertices (0, 0,
√
6
12 ,
√
10
4 ),
(0, 0,
√
2
3 , 0),
1√
3
(cos π3 , sin
π
3 , 0, 0),
1√
3
(cos 2π3 , sin
2π
3 , 0, 0) and
1√
3
(1, 0, 0, 0) is a Lenz
Configuration and that it is strictly configured. This is the only case with d = 4 where the
vertices of a strictly configured polytope is a Lenz Configuration.
Theorem 2.3. Let X ⊂ R4 be a 4-dimensional extremal Lenz Configuration with |X| ≥ 6.
Then P = conv(X) is not configured.
Proof. We assume X ⊂ R4 is a 4-dimensional Lenz Configuration with X ⊂ C1 ∪ C2,
Ci ⊂ Ui, where R4 = U1 ⊕ U2. It is clear that P is a 4-polytope with V (P ) = X and
diameter 1. Let X ∩ C1 = {w1, ..., wa}, X ∩ C2 = {z1, ..., zb} and note that for i = 1, 2,
Gi := Ui ∩ P ∈ F2(P ).
From [17], we have that M4(X) = M(4, n) with |X∩C1| = ⌈n2 ⌉ and |X∩C2| = ⌊
n
2 ⌋,
say. Furthermore, M(4, 6) = t2(6) + 4, M(4, 7) = t2(7) + 4 and M(4, n) ≤ t2(n) +
⌈n2 ⌉+ 1 for n ≥ 8 where t2(n) is the number of pairs {wj , zk} such that ∥wj − zk∥ = 1.
Accordingly, there are M(4, n) − t2(n) diameters of X that have end points in either C1
or C2.
We suppose that P is configured via the anti-isomorphism induced by v → v∗, v /∈ v∗,
and seek a contradiction. Then a ≥ 3, b ≥ 3, v /∈ v∗ and F(P ) = {w∗1 , ..., w∗a, z∗1 , ..., z∗b }
yield that v∗ ∩ C1 ̸= ∅ ̸= v∗ ∩ C2 for v ∈ X ∩ C1, and G1 = z∗1 ∩ z∗2 and G2 = w∗1 ∩ w∗2
say: Thus, w∗j ∩G1 ∈ F1(w∗j ) and z∗k ∩G2 ∈ F1(z∗k) for 3 ≤ j ≤ a and 3 ≤ k ≤ b.
It now follows that the number of principal diagonals of P in G1 and G2 is:
• two through each wj and zk with j ≥ 3, k ≥ 3 and
• at least one through each of w1, w2, z1 and z2;
that is, at least 12 (2(a−2)+2(b−2)+4) = a+b−2 = n−2 and n−2 ≤ ⌈
n
2 ⌉ ≠ 1. Then
n = 6, w∗3 ∩G1 = [w1, w2] and so, w3 ∈ w∗1 ∩ w∗2 , [G1, w3] ⊂ w∗1 ∩ w∗2 , and w∗1 = w∗2 ; a
contradiction.
We note that the arguments and the result in Theorem 2.3 extend to d ≥ 5 for extremal
Lenz Configuration X with sufficiently large |X|. This raises the issue of how to realize
configured polytopes with a large number of vertices in higher dimensions.
3 Constructions of strictly configured polytopes
In this section, we present realizations of strictly configured polytopes that are (d − 2)-
fold d-pyramids or “stratified” d-polytopes. We note that configured polytopes play an
important part in the study of, among others, graphs, hypergraphs, and bodies of constant
width.
T. Bisztriczky et al.: Configured polytopes and extremal configurations 641
3.1 Prismoids
Let m ≥ d ≥ 3 and Q ⊂ Hd(0) be a (d − 1)-polytope with V (Q) = {w1, w2, . . . , wm}
and cd as a relative interior point.
We consider translated homothetic copies (homotheties) Qjm of Q. For k ≥ 2 and
1 ≤ j ≤ k, let Qjm = [yj1, yj2, . . . , yjm] with yjr = (λjrwr, pj), pk < pk−1 < · · · < p1
and λj > 0. We let Rkm = [Q1m,Q2m, . . . ,Qkm], and say that Rkm is a k-layered
d-prismoid if |V (Rkm)| = km and for r = 1, . . . ,m, [y(j−1)r, yjr] are the edges of Rkm
that intersect Q(j−1)m and Qjm.
Then [Qim,Qjm] is a d-prismoid for 1 ≤ i ≤ j ≤ m, {Q1m,Qkm} ⊂ F(Rkm) and
we let Pkm = [y00, Rkm] for some point y00 = (0, . . . , 0, q) ∈ Rd. We say that Pkm is a
stratified d-polytope if y00 is beyond either Q1m or Qkm, and beneath all other facets of
Rkm (cf. [6] p. 78), and hence, |V (Pkm)| = km+ 1.
In what follows, we assume that Pkm = [y00, Rkm] ⊂ Rd is stratified with Rkm as
above and y00 beyond exactly Q1m. It is clear that Pkm is dependent upon the (d − 1)-
polytope Q = [w1, w2, . . . , wm] ⊂ Hd(0), and we examine properties of Pkm that are
inherited from Q.
As a point of reference, P2m ⊂ R3 is called an apexed 3-prism in [11].
3.1.1
Let Q = [w1, w2, . . . , wm] ⊂ Hd(0) be involutary self-dual via the anti-isomorphism on
L(Q) induced by wr → w̃r ∈ F(Q). Then F(Q) = {w̃1, w̃2, . . . , w̃m} and we have that
• Qjm is involutary self-dual via the anti-isomorphism of L(Qjm) that sends yjr →
ỹjr, and yjs ∈ ỹjr if, and only if, ws ∈ w̃r,
• F(Qjm) = {ỹj1, ỹj2, . . . , ỹjm},
• F(Rkm) = {Q1m,Qkm} ∪ {ỹ(j−1)r, ỹjr}|2 ≤ j ≤ k, 1 ≤ r ≤ m} and
• F(Pkm) = (F(Rkm) \ {Q1m}) ∪ {[y00, ỹ1r]|1 ≤ r ≤ m}.
Then (cf. [2], Theorem 2.1) Pkm is involutary self-dual via the anti-isomorphism on
L(Pkm) induced by the map yjr → Yjr with Y00 = Qkm, Ykr = [y00, ỹ1r] and Yjr =
[ỹ(k−j)r, ỹ(k−j+1)r] for j = 1, . . . , k − 1 and r = 1, . . . ,m. □
3.1.2
With Q as in 3.1.1, let V (Q) ⊂ Sd−2(cd, t) ⊂ Hd(0) and ∥wr − ws∥ = 1 for each
wr ∈ V (Q) and ws ∈ w̃r. We say that Pkm is metrically embedded in Rd if ∥y − y′∥ = 1
for every {y, y′} ⊂ V (Pkm) such that [y, y′] is a principal diagonal of Pkm. Thus, a
metrically embedded Pkm of diameter 1 is configured.
From Theorem 4.1 in [2]; if y00 = (0, 0, . . . , 0, q), then there are 0 < λk ≤ λ1 < · · · <
λj ≤ λk−j < · · · < λ[ k+12 ] = 1 that yield 0 = pk < pk−1 < · · · < p1 < q so that for every
yjr ∈ V (Pkm): if yis ∈ Yjr then ∥yjr−yis∥ = 1. Specifically, we note that q2 = 1−λ2kt2,
p21 = 1− ∥λkwr − λ1ws∥2 and pk−1 = p1 −
√
β with β = 1− ∥λk−1wr − λ1ws∥2. □
Our present interest is to determine involutary self-dual Pkm ⊂ Rd of, say, diameter 1
and then to characterize its diameters. To that end, we seek involutary self-dualQ ⊂ Hd(0)
of diameter 1 and with vertices on a (d− 2)-sphere.
642 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648
3.2 Pyramids with polygonal bases
With the ai’s to be specified, let d ≥ 3 and Q ⊂ L2(−a3, . . . ,−ad) be a regular m-gon
with cyclically labeled vertices w1, w2, . . . , wm, the circumradius t, the diameter 1 and
m = 2u + 1 ≥ 3. Then it is well known that 1 = ∥wr − wr+u∥ = ∥wr − wr+u+1∥ for
each wr, and that Q has 2m diameters.
As a simplification, we write wr = (x1, x2,−a3, . . . ,−ad) as wr = (x1, x2) in relation
to the plane L2(−a3, . . . ,−ad).
3.2.1
With θ = 2πm and wr = t(cos(rθ), sin(rθ)) for r = 1, . . . ,m, we note that wm =
(t, 0), wm+u = wu and 1 = ∥wm − wu∥2 = 2t2(1 − cos(uθ)) = 2t2(1 + cos( πm ))
from m = 2u+ 1.
3.2.2
With m = 2u + 1 ≥ 5 and λ > 0, we claim that ∥λwr − wj∥ < ∥λwr − wr+u∥ for
wj ∈ V (Q) \ {wr, wr+u, wr+u+1}.
With coordinates as in 3.2.1, we may assume that wr = wm and that wj is in the
upper half-plane. Then 0 < jθ < uθ < π and cos(uθ) < cos(jθ) and ∥λwm − wu∥2 −
∥λwm − wj∥2 = 2λt2(cos(jθ)− cos(uθ)). □
3.2.3
For λ > µ > 0 and ws ∈ {wr+u, wr+u+1}, we have that [λwr, µwr, µws, λws] is an
isosceles trapezoid of side lengths λ, µ and (λ−µ)t and ∥λwr−µws∥2 = λµ+(λ−µ)2t2 =
∥λws − µwr∥2. □
3.2.4
From 1 = ∥wm − wu∥2 = 2t2(1 + cos( πm )) and m ≥ 3, we obtain that
1
4 < t
2 ≤ 13 and
1
3 <
1
4(1−t2) ≤
3
8 . We let t2 = t, t
2
d =
1
4(1−t2d−1)
for d ≥ 3 and note that 13 < t
2
3 ≤ 38 <
t24 ≤ 25 < t
2
5 ≤ 512 < t
3
6 ≤ 37 < t
2
7 ≤ 716 < t
2
d <
1
2 with d ≥ 8. □
3.2.5
With d ≥ 4 and Q ⊂ L2(−a3, . . . ,−ad) ⊂ L3(−a4, . . . ,−ad) as above, we write wr =
(t2 cos(rθ), t2 sin(rθ),−a3) in relation to L3(−a4, . . . ,−ad). We consider the 2-sphere
S2 := S2 ((0, 0, 0), t3) ⊂ L3(−a4, . . . ,−ad) with t23 = 14(1−t22) , and let a3 =
√
t23 − t22.
Then V (Q) ⊂ S2 and with wm+1 = (0, 0, t3), we claim that ∥wm+1 − wr∥ = 1 for
r = 1, 2, . . . ,m.
AsQ is symmetric about the x3-axis, we verify the claim with wr = wm = (t2, 0,−a3).
From t23 = ∥wm∥2 = t22 + a23 and t22 =
4t23−1
4t33
, it follows that ∥wm+1 − wm∥2 =
t22 + (t3 + a3)
2 = 2t23 + 2t3
√
t23 − t22 = 2t23 + 2t3
(
(1−2t23)
2
4t23
) 1
2
= 1.
T. Bisztriczky et al.: Configured polytopes and extremal configurations 643
Theorem 3.1. Let d ≥ 3 and Q2 = [w1, . . . , wm] ⊂ L2(−a3, . . . ,−ad) be a regular
m-gon of diameter 1 and circumradius t2; m = 2u + 1 ≥ 3. Then for e = 3, . . . , d,
t2e =
1
4(1−t2e−1)
, a2e = t
2
e − t2e−1 and ce = (0, . . . ,−ae+1, . . . ,−ad) if e ̸= d, there is an
involutary self-dual (e− 2)-fold e-pyramid Qe = [w1, . . . , wm, . . . , wm+e−2] of diameter
1 and basis Q2 such that
(i) Qe ⊂ Le(−ae+1, . . . ,−ad) if e ̸= d,
(ii) V (Qe) ⊂ Se−1(ce, te) and
(iii) Qe is strictly configured.
Proof. With reference to Subsections 3.2.1, 3.2.2, 3.2.3, 3.2.4 and 3.2.5, we let:
• wi = (t2 cos(iθ), t2 sin(iθ),−a3, . . . ,−ad) for i = 1, . . . ,m
• wm+i = (0, . . . , 0, ti+2,−ai+3, . . . ,−ad) for i = 1, . . . , d− 3 and
• wm+d−2 = (0, . . . , 0, td).
We observe first that for 2 ≤ i < j ≤ d, t2i +a2i+1 = t2i+1 and so, t2i +a2a+1+ · · ·+a2j = t2j .
From this it follows that ∥wi − ce∥2 = t22 + a23 + · · · + a2e = t2e for wi ∈ V (Q2),
3 ≤ e ≤ d ∥wm+i − ce∥2 = t2i+2 + a2i+3 + · · · + a2e = t2e for i + 2 ≤ e ≤ d − 1 and
∥wj − cd∥2 = ∥wj∥2 = t2d for wj ∈ V (Qd).
Next, with wr = (t2 cos(rθ), t2 sin(rθ),−a3, . . . ,−ad) and w′r = (t2 cos(r + u)θ,
t2 sin(r + u)θ,−a3, · · · − ad), we note that Q2 is involutary self-dual via the
anti-isomorphism of L(Q2) induced by wr → w̄r = [w′r, w′r+1]. Then for e = 3, . . . , d,
F(Qe) = {[w̄r, wm+1, . . . wm+e−2]|r = 1, . . .m}∪{[V (Qe)\{wr}]|r = m+1, . . . ,
m+ e− 2}
andQe is involutary self-dual via the anti-isomorphism onL(Qe) induced by wr → w̃r
where
w̃r =
{
[w̄r, wm+1, . . . wm+e−2], r = 1, . . . ,m;
[V (Qe) \ {wr}], r = m+ 1, . . . ,m+ e− 2.
Finally, we observe that for 1 ≤ j ≤ m+ i, ∥wm+i − wj∥2 = t2i+1 + (ti+2 + ai+2)2.
Then, as in 3.2.5, t2i+1 =
4t2i+2−1
4t2 yields that ∥wm+i − wj∥ = 1. From this and t
2
2 =
1
2(1+cos( πm )
, we obtain that ∥wr − ws∥ = 1 for ws ∈ w̃r; furthermore, if {wr, wz} ⊂
V (Q2) and wz /∈ w̃r then ∥wr − wz∥ < 1.
We note that Me(Qe) = 2M2(Q2) +
∑m+e−3
m+1 j = (e − 1)m +
(
e−2
2
)
and that Q3 is
extremal.
Theorem 3.2. Let d ≥ 3, m = 2u + 1, n = m + d − 3 and k ∈ {2, 3}. Then there
is an involutary self-dual stratified Pkn = [y00, Rkn] ⊂ Rd of diameter 1 that is strictly
configured.
Proof. With reference to Subsection 3.1 and Theorem 3.1 with e = d− 1 and ad = 0, we
consider Pkn with the property that:
• y00 is beyond exactly Q1n.
644 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648
• Q = [w1, ..., wn] ⊂ Ld−1(−ad) = Hd(0),
• Qd−1 is a involutary self-dual (d−3)-fold (d−1)-pyramid with diameter 1 and basis
Q2, and
• Q2 = [w1, ..., wm] ⊂ L2(−a3, ...,−ad) is a regular m-gon of diameter 1.
Then cd−1 = (0, . . . , 0,−ad) = cd and with t2, . . . td−1 as in 3.2.4, we simplify notation
and let t = td−1.
We now apply 3.1.2 with y00 = (0, . . . , 0, q) and pk < pk−1 < · · · < p1 < q.
Case 1: k = 2 and hence, λ1 = 1 and p2 = 0.
With 0 < λ2 < 1: P2n is stratified, Y00 = Q2n, Y1r = [ỹ1r, ỹ2r] and Y2r = [y00, ỹ1r].
With q2 = 1− λ2t2 and p21 = 1−∥λ2wr −ws∥2 = 1− (λ2 + (1− λ2)2t2) (cf. 3.2.3), we
have that ∥yjr − yis∥ = 1 for yis ∈ Yjr.
With λ2 = 12 ; we have q
2 = 4−t
2
4 , p
2
1 =
2−t2
4 and claim that ∥yjr − yiz∥ < 1 for
yiz /∈ Yjr. From 13 < t
2 < 12 , we obtain that
∥y00 − y1r∥2 = ∥(0, q)− (wr, p1)∥2 = ∥wr∥2 + (q − p1)2
= t2 + q2 + p21 − 2qp1
=
1
4
(6− 2t2 − 2
√
4− t2
√
2− t2)
≤ 1
4
(
6 + 2
(
1
2
)
− 2
√
4− 1
3
√
2− 1
3
)
< 1
(3.1)
Let yiz ̸= y00 ̸= yjr and yiz /∈ Yjr. Then yiz = (λiwz, pi), yjr = (λjwr, pj) and wz /∈
w̃r (cf. 3.1.1). Since Q1n and Q2n are homothets of Q, we may assume by Theorem 3.1(iii)
that j = 1 and i = 2, say. Since wz /∈ w̃r, it follows as in the proof of Theorem 3.1 that
wz = wr or {wz, wr} ⊂ V (Q2). If wz = wr, then ∥y1r − y2r∥2 = t
2
4 + p
2
1 =
1
2 . If
{wz, wr} ⊂ V (Q2), then it follows from 3.2.2 that ∥wr − 12wz∥ < ∥wr −
1
2ws∥ with
ws ∈ w̃r ∩ V (Q2). Hence, ∥y1r − y2z∥ < ∥y1r − y2s∥ = 1.
Case 2: k = 3 and hence, λ2 = 1 and p3 = 0.
Let Y00 = Q3n, Y1r = [ỹ2r, ỹ3r], Y2r = [ỹ1r, ỹ2r] and Y3r = [y00, ỹ1r]. With
λ = λ1 = λ3 =
1
2 and q
2 = 1−λt2 = 4−t4 , p
2
1 = 1−∥λwr−λws∥2 = 1−λ2 = 34 (cf. 3.1.2
and 3.2.3), β = 1− ∥λ2wr − λ1ws∥2 = 1− ∥wr − λws∥2 = 1− λ+ (1− λ)2t2 = 2−t
2
4
and p2 = p1 −
√
β, we obtain that ∥yjr − yis∥ = 1 for yis ∈ Yjr.
Let yiz /∈ Yjr. We claim that ∥yjr − yiz∥ < 1 and then it follows that each Yjr is a
facet of P3n; that is, R3n is a 3-layered prismoid and P3n is stratified.
We observe that if a < t2 ≤ b then
∥y00 − y2r∥2 = ∥(0, q)− (wr, p2)∥2 = ∥wr∥2 + (q − p2)2
= t2 + q2 + p21 + β + 2q
√
β − 2p1
(
q +
√
β
)
=
1
4
(
9 + 2t2 + 2
√
(4− t2)(2− t2)− 2
√
3(
√
4− t2 +
√
2− t2
)
<
1
4
(
9 + 2b+ 2
√
(4− a)(2− a)− 2
√
3(
√
4− b+
√
2− b
)
(3.2)
T. Bisztriczky et al.: Configured polytopes and extremal configurations 645
and ∥y00 − y2r∥ < 1 for (a, b) ∈ {( 13 ,
3
8 ), (
3
8 ,
2
5 ), (
2
5 ,
5
12 ), (
5
12 ,
3
7 ), (
3
7 ,
7
16 ), (
7
16 ,
1
2 )}, that
is, for each d ≥ 3 (cf. 3.2.4).
It is clear that ∥y00 − y1r∥ < ∥y00 − y2r∥, and hence, we may assume that
yiz = (λiwz, pi), yjr = (λjwr, pj) and wz /∈ w̃r. Then ∥wr − wz∥ < ∥wr − ws∥
for ws /∈ w̃r, and ∥y1r − y3z∥ < ∥y1r − y3s∥ = 1 for y3s ∈ ỹ1r ⊂ Y1r.
From t2 < 12 , we obtain that β >
3
16 =
p21
4 , p2 = p1 −
√
β < p12 and p2 < p1 − p2.
Thus, ∥y3r − y2z∥ < ∥y1r − y2z∥ and we argue as above that ∥y1r − y2z∥ < 1.
In summary; ∥yjr − y′∥ < 1 for {yjr, y′} ⊂ {y00} ∪ {yjr|j = 1, . . . , k and
r = 1, . . . , n}, and with equality if and only if y′ ∈ Yjr. Thus
F(Pkn) = {Y00} ∪ {Yjr|j = 1, . . . , k, r = 1, . . . , n},
V (Pkn) = {y00} ∪ {yjr|j = 1, . . . , k, r = 1, . . . , n}
and Pkn is involutary self-dual under the anti-isomorphism on L(Pkn) induced by yjr →
Yjr.
Theorem 3.3. Let Pkm ⊂ R3 be an involutary self-dual stratified 3-polytope that is con-
figured with diameter 1; k ≥ 2 and m = 2u+ 1 ≥ 3. Then there is an involutary self-dual
stratified P(k+1)m ⊂ R3 that is configured with diameter 1.
Proof. We let l = k + 1 and denote Pkm as in 3.1.1 and 3.1.2 with d = 3. Specifically,
• Q = [w1, . . . , wm] ⊂ H3(0) is a regular m-gon of diameter 1 and circumcentre
c3 = (0, 0, 0) as in 3.2.1,
• Qjm = [yj1, . . . , yjm] with yjr = (λjwr, pj) and 0 < λk ≤ λ1 < · · · < λj ≤
λk−j < · · · < λ[ l2 ] = 1, 0 < pk < pk−1 < · · · < p1 < q ≤ 1 and y00 = (0, 0, q),
• the anti-isomorphism on L(Pkm) is induced by yjr → Yjr with Y00 = Qkm, Ykm =
[y00, ỹ1r], Yjr = [ỹ(k−j)r, ỹ(l−j)r], 1 ≤ j ≤ k − 1, and ỹjr = [yj(r+u), yj(r+u+1)],
and
• ∥yjr − yis∥ = 1 if, and only if, yis ∈ Yjr.
Let S(y) := S2(y, 1) for y ∈ R3, and consider the homothets Q0m = [y01, . . . , yom] of
Q with y0r = (λ0wr, p0), 0 < λ0 < λ1 and p1 < p0 < q. From [yk(r+u), yk(r+u+1)] =
ỹkr = Y00∩Y1r, it follows that ∥y00−yks∥ = 1 = ∥y1r−yrs∥ for s ∈ {r+u, r+u+1},
and so,
{y00, y1r} ⊂ Ckr := S(yk(r+u)) ∩ S(yk(r+u+1)),
a circle with centre 12 (yk(r+u) + yk(r+u+1)). It is now clear that
(i) for each p1 < p0 < q, there is 0 < λ0 < λ1 such that y0r ∈ Ckr.
In fact, y0r ∈ αkr, the shorter arc of Ckr with end points y00 and y1r. We note also
that V (Q0m) ∩ V (Pkm) = ∅ for each such p0. Let V = V (Pkm), B(y) = [S(y)]
and B(V ) = ∩y∈V B(y). Since diam(Pkm) = 1, it follows that
(ii) αkr ⊂ bd(B(V )) for r = 1, . . .m.
Since Pkm is involutary self-dual with no fixed points, it follows from Theorem 3.2
of [13] that B(V ) is polytopal and the face polyhedral structure of B(V ) is a lattice
646 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648
isomorphic to L(Pkm). Accordingly, B(V ) is similarly self-dual and from Theo-
rem 4.1 of [13], any surface Φ ⊂ R3 obtained from bd(B(V )) (by performing their
surgery on one edge-arc of each pair of dual edge-arcs of bd(B(V ))) is the boundary
of a body of constant width. In this case, V ⊂ Φ and diam(V ) = 1 yield Φ is of
constant width 1.
We note that dual edge-arcs of bd(B(V )) correspond to dual edges of L(Pkm).
Thus, the duality [y00, y1r]←→ Y00 ∩ Y1r = ỹkr
yields that αkr is dual to the shorter edge-arc in S(y00) ∩ S(y1r) with end point
yk(r+u) and yk(r+u+1). We consider those Φ that contain each of αk1, αk2, . . . , αkm.
Then the symmetry of Pkm about the x3–axis and i) yield that
(iii) V ′ = V ∪ V (Q0m) ⊂ Φ and diam(V ′) = 1,
(iv) S(y00)∩V ′ = V (Qkm) and the spherical region S(y00)∩Φ is not empty and bounded
in H3(0) by the circumcircle of Qkm, and
(v) y′00 = (0, 0, q − 1) ∈ S(y00) ∩ Φ.
From diam(V ) = 1, |V | = km + 1, M(3, km + 1) = 2km and Theorem 2.1, we
have that M3(V ) = 2km+ 1. From diam(V ′) = 1, |V ′| = lm+ 1 and i), we have
that M3(V ′) ≥M3(V ) + 2m = 2lm. Thus, M3(V ′) = 2lm and
(vi) ∥y0r − y∥ < 1 for y0r ∈ V (Q0m) and y ∈ V \ {yk(r+u), yk(r+u+1)}.
Let V ′′ = V ′ ∪ {y′00}. Then diam(V ′′) = 1, |V ′′| = lm+ 2, ∥y00 − y′00∥ = 1 and
2|V ′′| − 2 = 2lm+ 2 ≥M3(V ′′) ≥ 2lm+ 1. From the rotational symmetry of V ′′
and S(y′00) about the x3–axis, it follows that
(vii) ∥y′00 − y∥ < 1 for y ∈ V ′ \ {y0}, and
(viii) ∥yϵ−y∥ < 1 for y ∈ V ′ \{y0} for sufficiently small ϵ > 0 and yϵ = (0, 0, q−1−ϵ).
Let p0 = q−ϵ and µ be the radius of the circle H3(p0)∩S(y′00). Then {(0, 0, p0)} =
H3(p0)∩S(yϵ) ⊂ Q0m ⊂ [H3(p0)∩S(y′00)] and with λ0 chosen so that 0 < λ0 < λ1
and y0r ∈ αkr, we have that 0 < λ0t ≤ µ. Accordingly, there is a point z00 ∈
[y′00, yϵ] such that λ0t is the radius of H3(p0) ∩ S(z00); that is,
(ix) ∥z00 − y0r∥ = 1 for r = 1, 2, . . . ,m.
Finally, let zjr = y(l−j)r,z̃jr = ỹ(l−j)r and Q′jm = Q(l−j)m for j = 1, 2, . . . , l and
r = 1, 2, . . . ,m. In addition, let Z00 = Q′lm = Q0m, Zlr = [z00, z̃1r] = [z00, ỹkr]
and Zjr = [z̃(l−j)r, z̃(l−j+1)r] = [ỹjr, ỹ(j−1)r]. From the preceding, we have that
Plm = [z00, Q
′
1m, . . . , Q
′
lm] is involutary self-dual via zjr → Zjr, stratified and
configured with diameter 1.
Finally, we show that if a set of n points are the vertices of a configured 4-polytope P
such as in Theorem 3.2 then M4(P ) ≤ 4n.
Theorem 3.4. Let Pkm = [y00, Rkm] ⊂ R4 be a configured stratified 4-polytope, with
n = km+ 1 vertices. Then number of principal diagonals of Pkm is at most 4n.
Proof. By Theorem 2.2, it is sufficient to prove that f1(P ) ≤ 3n for every configured
stratified 4-polytope. By construction, Rkm = [Q1m,Q2m, . . .Qkm] where each copy
T. Bisztriczky et al.: Configured polytopes and extremal configurations 647
Qim is self-dual and contains m vertices, and thus, f1(Qim) = 2m−2 by Euler’s Theorem
and self-duality.
Finally, there are m edges through y00 and m(k−1) edges connecting the k homothets
Qim, and so, f1(Pkm) = k(2m− 2)+m(k− 1)+m = 3km− 2k ≤ 3km+3 = 3n.
ORCID iDs
Tibor Bisztriczky https://orcid.org/0000-0001-7949-4338
Gyivan Lopez-Campos https://orcid.org/0000-0003-2005-8210
Deborah Oliveros https://orcid.org/0000-0002-3330-3230
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