Selfishness and Rupert Property of convex bodies Liping Yuan - - PowerPoint PPT Presentation

Selfishness and Rupert Property

• f convex bodies

Liping Yuan College of Mathematics and Information Science Hebei Normal University Shijiazhuang, China Shanghai Jiaotong University

Part I. Selfishness of Convex Bodies

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 2 / 67

F-convexity

Tudor Zamfirescu proposed at the 1974 meeting on Convexity in Oberwolfach the investigation of F-convexity, for various families F. Let F be a family of sets in Rd. A set M ⊂ Rd is called F-convex1 if for any pair of distinct points x, y ∈ M there is a set F ∈ F such that x, y ∈ F and F ⊂ M.

F M

x y

F ∈ F and x, y ∈ F ⊂ M

1Blind R, Valette G, Zamfirescu T. Rectangular convexity [J]. Geom.

Dedicata, 1980, 9: 317-327.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 3 / 67

F-convexity members of F the usual convexity usual closed line-segments affine linearity lines arc-wise connectedness arcs polygonal connectedness polygonal paths

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 4 / 67

F is a family of connected sets

Authors Members of F F-convexity Blind2 and B¨

• r¨
• czky3

all non-degenerate rectangles r-convexity Bruckner4 and Magazanik5 all polygonal paths in the plane with length at most n Ln-convexity

2Blind R, Valette G, Zamfirescu T. Rectangular convexity [J]. Geom.

Dedicata, 1980, 9: 317-327.

3B¨

• r¨
• czky K Jr. Rectangular convexity of convex domains of constant width

[J]. Geom. Dedicata, 1990, 34: 13-18.

4Bruckner A M, Bruckner J B. Generalized convex kernels[J]. Israel J.

Math., 1964, 2(1): 27-32.

5Magazanik E, Perles M A. Generalized convex kernels of simply connected

Ln sets in the plane[J]. Israel J. Math., 2007, 160(1): 157-171.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 5 / 67

F is a family of connected sets

Authors Members of F F-convexity Magazanik et al.6 all polygonal paths in R2 with sides parallel to the coordinate axis staircase convexity Zamfirescu7 right triangles in a Hilbert space of dimension at least 2 right convexity

6Magazanik E, Perles M A. Staircase connected sets[J]. Discrete Comp.

Geom., 2007, 37: 587-599.

7Zamfirescu T. Right convexity [J]. J. Convex Anal., 2014, 21: 253-260. Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 6 / 67

F is a family of discrete point sets

Authors Members of F F-convexity Yuan et al.8 9 all triples {x, y, z} with ∠xyz = π

2

rt-convexity Yuan et al.10 all triples whose convex hull are isosceles triangles (maybe degenerated triangles) it-convexity Li et al.11 all quadruple whose convex hull is a non-degenerate rectangle rq-convexity

8Yuan L, Zamfirescu T. Right triple convex completion [J]. J. Convex Anal.,

2015, 22(1): 291-301.

9Yuan L, Zamfirescu T. Right triple convexity [J]. J. Convex Anal., 2016, 23:

1219-1246.

10Yuan L, Zamfirescu T, Zhang Y. Isosceles triple convexity [J]. Carpathian J.

Math., 2017, 33(1): 127-139.

11Li D, Yuan L, Zamfirescu T. Right quadruple convexity [J]. Ars Math.

Contemp., 2018, 14: 25-38.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 7 / 67

For a subset of the vertex set of a graph, the g-convexity investigated by Farber and Jamison12, the T-convexity studied by Changat and Mathew13, and M-convexity researched by Duchet14 can also be regarded as examples of F-convexity for suitable families F.

• 12M. Farber, R. E. Jamison. On local convexity in graphs[J]. Discrete Math.,

66 (1987) 231-247.

• 13M. Changat, J. Mathew. On triangle path convexity in graphs, Discrete

Math., 206 (1999) 91-95.

• 14P. Duchet. Convex sets in graph II. Minimal path convexity, Combin.

Theory Ser. B, 44 (1988) 307-316.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 8 / 67

Selfishness of compact sets

A family F of compact sets is complete if F contains all compact F-convex sets.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 9 / 67

Selfishness of compact sets

A family F of compact sets is complete if F contains all compact F-convex sets. A compact set K is called selfish, if the family FK of all compact sets similar to K is complete.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 9 / 67

Example: Every square is selfish

Let Q be a square, and let K be a compact FQ-convex set. We show that K is also a square.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 10 / 67

Notation

As usual, for M ⊂ Rd with d ≥ 2, clM denotes its topological closure, bd M its boundary, and diam M = supx,y∈M x − y. A 2-point set {x, y} ⊂ M with x − y = diam M is called a diametral pair of M, while xy is a diameter of M. For any compact set C ⊂ Rd, let SC be the smallest hypersphere containing C in its convex hull. Let K be the space of all convex bodies in Rd, endowed with the Pompeiu-Hausdorff metric. A convex body K is called long if card(K ∩ SK) = 2. The unit ball is denoted by B, and bd B = S.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 11 / 67

Non-selfish convex bodies

Unimodal

A continuous real function defined on an interval [a, b] is unimodal if it is non-decreasing on a subinterval [a, c] and non-increasing on [c, b]. A C2-arc A in R2 will be called here unimodal, if its curvature radius at x ∈ A is a unimodal function of arc-length (from an endpoint of A to x). Theorem: Suppose K ⊂ R2 is a long convex body. If at least one of the arcs in bd K between the two points of K ∩ SK is unimodal, then K is not selfish.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 12 / 67

The condition card(K ∩ SK) = 2 alone does not guarantee non-selfishness. Not every polygon is selfish either. For instance, if an edge ab of the polygon P is a diameter of SP , then the disc is FP -convex, as one easily verifies. So, among the triangles, all non-acute ones are non-selfish. Are all acute ones selfish?

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 13 / 67

Selfishness of triangles

Theorem: The equilateral triangle is selfish.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 14 / 67

Is every acute triangle is selfish ?

Theorem: There exist non-selfish acute triangles. We prove that the triangle x0dz0 is not selfish, by showing that abcde is Fx0dz0-convex.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 15 / 67

Selfishness of triangles

Theorem: Every isosceles acute triangle is selfish.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 16 / 67

Selfishness of quadrilaterals

Regarding families F of sets, an interesting case is that of all

• rectangles. The family F is not complete, but a characterization of

F-convexity in the compact case is still missing. Theorem: Every rectangle is selfish in the plane.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 17 / 67

Selfishness of quadrilaterals

In the case of the rhombus, while every rhombus is selfish, the family R of all of them is not complete, as again the disc demonstrates. Theorem: Every rhombus in R2 is selfish.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 18 / 67

Selfishness of regular convex polygons

Theorem: Every regular convex polygon is selfish.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 19 / 67

Other non-selfish convex bodies

Concerning the selfishness of long polytopes, it is clear that a necessary condition is that their diameter is not contained in a face. Is this also sufficient? We have seen, indeed, that every rhombus is selfish. Theorem: There exists a long polygon with its diameter not among the sides, which is not selfish.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 20 / 67

Other non-selfish convex bodies

As all rectangles are selfish in R2, we might be inclined to think that also (some) circular cylinders in R3 are selfish...

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 21 / 67

Other non-selfish convex bodies

As all rectangles are selfish in R2, we might be inclined to think that also (some) circular cylinders in R3 are selfish... Theorem: No circular cylinder in R3 is selfish.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 21 / 67

Families of convex bodies and selfishness

The convex bodies of constant width are selfish and the family of all

• f them complete.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 22 / 67

Families of convex bodies and selfishness

The convex bodies of constant width are selfish and the family of all

• f them complete.

Theorem: If the families of sets {Fι}ι∈I are complete, then

ι∈I Fι

is also complete.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 22 / 67

Families of convex bodies and selfishness

The convex bodies of constant width are selfish and the family of all

• f them complete.

Theorem: If the families of sets {Fι}ι∈I are complete, then

ι∈I Fι

is also complete. Theorem: There exist selfish convex bodies K, K′, such that FK ∪ FK′ is not complete.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 22 / 67

Families of convex bodies and selfishness

The convex bodies of constant width are selfish and the family of all

• f them complete.

Theorem: If the families of sets {Fι}ι∈I are complete, then

ι∈I Fι

is also complete. Theorem: There exist selfish convex bodies K, K′, such that FK ∪ FK′ is not complete. Theorem: Let F be a family of sets and G a family of F-convex sets. If some set is G-convex, then it is also F-convex.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 22 / 67

For most convex bodies K in Rd (in the sense of Baire categories), card(K ∩ SK) = d + 1. Moreover, for all convex bodies in Rd, except those in a nowhere dense subset, card(K ∩ SK) ≥ d + 1. Hence, the set of long convex bodies is small in K. However, since the set K′ of all convex bodies K, for which K ∩ SK contains a diametral pair of SK, is closed in K, it is itself a Baire space. Most of these convex bodies are long. We want to look for the chances of a convex body in K′ to be selfish.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 23 / 67

Families of convex bodies and selfishness

Theorem: For all pairs of convex bodies (K, K′) ∈ K × K except those of a nowhere dense family of pairs, K is not FK′-convex.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 24 / 67

Families of convex bodies and selfishness

Theorem: For all pairs of convex bodies (K, K′) ∈ K × K except those of a nowhere dense family of pairs, K is not FK′-convex. Theorem: For every smooth K′ ∈ K, and for all convex bodies K ∈ K except those of a nowhere dense family, K is not FK′-convex. Considering K′ instead of K does not change the situation, and theorems analogous to above are true with K′ instead of K. So, the chances of a long convex body in K′, and even of a convex body in K, to be selfish are not bad!

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 24 / 67

Families of convex bodies and selfishness

Let F(K) be the set of all compact FK-convex sets. There are convex bodies K with huge F(K). However, F(K) cannot become K.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 25 / 67

Families of convex bodies and selfishness

Let F(K) be the set of all compact FK-convex sets. There are convex bodies K with huge F(K). However, F(K) cannot become K. Theorem: For each K ∈ K there exists K′ / ∈ F(K) such that K ∈ F(K′).

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 25 / 67

Selfishness of discrete point sets

Theorem The vertex set of a regular polygon is selfish.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 26 / 67

Selfishness of discrete point sets

Theorem The vertex set of an isosceles triangle, except the equilateral one, is not selfish.

• Figure: {a, b, c}, {a, d, e}, {b, c, e}, {c, b, d}

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 27 / 67

Selfishness of discrete point sets

Theorem There exist a parallelogram whose vertex set is non-selfish.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 28 / 67

Conclusion

Convex bodies Selfishness of Selfishness of vertex sets convex polygons regular polygons selfish selfish non-equilateral isosceles non-selfish selfish acute triangles isosceles non-acute triangles non-selfish non-selfish rectangles with length ratio of non-selfish selfish long and short edges is √ 2 : 1

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 29 / 67

Part II. Rupert Property of Convex Bodies

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 30 / 67

Origin

More than three hundred years ago, Prince Rupert (1619-1682) won the wager that whether a hole large enough can be cut in one of two same cubes to permit another to pass through. 15

• 15J. Wallis, De Algebra Tractatus, 1685, in Opera Mathematica, vol. 2,

Oxoniae: E Theatro Sheldoniano, 1693, pp. 470-471.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 31 / 67

Origin

More than three hundred years ago, Prince Rupert (1619-1682) won the wager that whether a hole large enough can be cut in one of two same cubes to permit another to pass through. 15 This wager’s victory gave rise to the consideration of whether other convex bodies have this property.

• 15J. Wallis, De Algebra Tractatus, 1685, in Opera Mathematica, vol. 2,

Oxoniae: E Theatro Sheldoniano, 1693, pp. 470-471.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 31 / 67

A “hole” can be cut.

A hole is a straight tunnel.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 32 / 67

A “hole” can be cut.

A hole is a straight tunnel. πn: a plane with unit normal vector n. Pn: the projection map of R3 onto πn. γ: a simple closed curve that lies in the plane πn. Iγ: the domain in πn interior to τ. The hole Hγ with directrix γ and direction n is the set Hγ = {y + tn ∈ R3: y ∈ Iγ, −∞ < t < ∞}.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 32 / 67

Rupert property

A convex body P has the Rupert property, means that there are vectors n, m and an isometry µ of πn onto πm such that µ(Pn(P)) ⊂ int Pm(P). We say that P passes through the hole Hτ with directrix τ = bd Pm(P) and direction m.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 33 / 67

Rupert property

A convex body P has the Rupert property, means that there are vectors n, m and an isometry µ of πn onto πm such that µ(Pn(P)) ⊂ int Pm(P). We say that P passes through the hole Hτ with directrix τ = bd Pm(P) and direction m. Pn(P) is the inner projection of P, denoted by Pi; Pm(P) is the outer projection of P, denoted by Po.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 33 / 67

Convex bodies without Rupert property

The unit ball The equilateral drum (a circular cylinder of unit diameter and height closed on each end by disks)

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 34 / 67

Niewland constant

It is natural to ask how large a polytope P′ similar to P can be to pass through a hole in P, i.e., how large can a positive scalar ν be, such that the polytope νP passes through a suitable hole in P? We call this Nieuwland’s question after P. Nieuwland (1764õ1794), who asked and answered this question for the cube. Define the Nieuwland constant ν(P) of the polytope P by ν(P) = sup {ν > 0 : νP can pass through a suitable hole in P}.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 35 / 67

Platonic solids

Platonic Solid Nieuwland Constants Estimate Cube C16 ν(C) = 3

4

√ 2 ≥ 1.060 660 Tetrahedron T17 ν(T) ≥ 2

5

√ 3( √ 6 − 1) > 1.004 235 Octahedron O3 ν(O) ≥ 3

4

√ 2 ≥ 1.060 660 Dodecahedron D18 ν(D) ≥ 171

170 > 1.005 882

Icosahedron I4 ν(I) ≥ 1108

1098 > 1.009 107

• 16J. H. van Swinden, Grondbeginsels der Meetkunde (Elements of Geometry),

Pieter den Hengst and Son, Amsterdam, 1816.

17Christoph J. Scriba, Das Problem des Prinzen Ruprecht von der Pfalz,

Praxisder Math., 10 (9) (1968) 241-246.

• 18R. P. Jerrard, J. E. Wetzel, L. Yuan, Platonic Passages. Math. Mag., 90

(2017) 87-98.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 36 / 67

Universal stoppers

In 2008, Jerrard and Wetzel19 proved each universal stopper has the Rupert property.

19Richard P. Jerrard and John E. Wetzel, Multipurpose stoppers are Rupert,

• College. Math. J., 39 (2008) 90-94.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 37 / 67

Archimedean solids

An Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 38 / 67

Preliminaries

Now let ex = (1, 0, 0), ey = (0, 1, 0), ez = (0, 0, 1) be the standard basis for R3. And let Πxy be the plane spanned by ex, ey, the

• riginal x-axis be the new x-axis, and the original y-axis be the new y-axis.

Thus, Pez denotes the orthogonal projection of R3 onto Πxy. Let Tx, Ty, Tz denote the rotational transformations of R3 around x, y, z-axis by an angle α, β, γ, respectively. The rotation angle is positive if and only if the rotation obeys the right-hand rule.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 39 / 67

Preliminaries

P (x(α), y(β), z(γ)) means that P is rotated about the x-axis by an angle α, then about the y-axis by an angle β, and then about the z-axis by an angle γ. By the definition of the Rupert property, we only need to find two Pj = P(x(αj), y(βj), z(γj)) (j = 1, 2), that satisfy Pez(P1) ⊂ Pez(P2). We have Pez(P1) = Pi and Pez(P2) = Po.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 40 / 67

Cuboctahedron

Let C denote a cuboctahedron of edge length √ 2.

• The cuboctahedron C

The coordinates of C’s vertices

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 41 / 67

Cuboctahedron

First we consider the projection of C along lom. Now we rotate C by an angle − π

4 about the z-axis, and then by an

angle − arcsin

√ 6 3 about the x-axis. The cuboctahedron obtained is

denoted by C(z(− π

4 ), x(− arcsin √ 6 3 )).

• After rotation, the vector −

→

• m coincides with the z-axis. −

→

• m is the

direction vector of lom, so the projection of C(z(− π

4 ), x(− arcsin √ 6 3 ))

• nto Πxy is the same as the projection of C along lom. Take Po to be this

projection.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 42 / 67

Cuboctahedron

To find the inner projection Pi, we consider the projection of C along loa3. Rotate C about y-axis by π

4 ; the new cuboctahedron is denoted by

C(y( π

4 )).

• After rotation, the direction vector of loa3, −

→

• a3, coincides with the

z-axis. Therefore the projection of C(y( π

4 )) onto Πxy is the same as the

projection of C along loa3.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 43 / 67

Cuboctahedron

Let Po and C(y(− π

4 ))z be projections of C(z(− π 4 ), x(− arcsin √ 6 3 ))

and C(y( π

4 )) onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 44 / 67

Cuboctahedron

If C(y( π

4 )) is rotated by an angle β (> 0) about the y-axis, the vertex

2z will move closer to the point p along the line lpq, and the vertex 1z will move closer to the point o along the x-axis.

• Then we choose a suitable β, such that 1′

z, 2′ z move into the interior

• f Po and 1′

z2′ z is parallel to 2z5z.

Because 1′

z2′ z is parallel to 2z5z if and only if 2′ z − p = 1′ z − 5z,

we get β = arccos √ 6 + 2 √ 3 6 ≈ 9.73561◦.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 45 / 67

Cuboctahedron

Po is the projection of C(z(− π

4 ), x(− arcsin √ 6 3 )) onto Πxy.

Pi is the projection of C(y( π

4 ), y(arccos √ 6+2 √ 3 6

)) onto Πxy.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 46 / 67

Icosidodecahedron

Let I be an icosidodecahedron with unit edge-length. Po is the projection of I(y(− arcsin

√ 5+1 2√ 2 √ 5+5)) onto Πxy.

Pi is the projection of I(y( π

9 ), x( π 180)) onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 47 / 67

Truncated icosahedron

Let J be a truncated icosahedron with edge-length 2. Po is the projection of J (y(arcsin √ 3 (

√ 5 6 − 1 6))) onto Πxy.

Pi is the projection of J (x( 23π

180 ), y( 2π 15 ), z( π 18)) onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 48 / 67

Truncated octahedron

Let O be a truncated octahedron with edge-length 2. Po is the projection of O(z(− π

4 ), x(− arcsin 1 3)) onto Πxy.

Pi is the projection of O(y(− π

4 ), z( 3π 10 )) onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 49 / 67

Truncated cube

Let T be a truncated cube with unit edge-length. Po is the projection of T (z( π

4 ), x(arcsin √ 6 3 )) onto Πxy.

Pi is the projection of T onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 50 / 67

Rhombicuboctahedron

Let R be a rhombicuboctahedron with edge-length 2. Po is the projection of R(z(arcsin 2

√ 5 5 ), x(arcsin √ 5

√

8 √ 2+17)) onto

Πxy. Pi is the projection of R(z( π

12)) onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 51 / 67

Truncated cuboctahedron

Let U be a truncated cuboctahedron with edge-length 2. Po is the projection of U(z( π

4 ), x(arcsin √ 2+ 3

2

• 6

√ 2+ 39

4

)) onto Πxy. Pi is the projection of U(z( π

20)) onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 52 / 67

Truncated dodecahedron

Let D be a truncated dodecahedron with edge-length √ 5 − 1. Po is the projection of D(y(− arcsin

√ 5+1

√

34 √ 5+90)) onto Πxy.

Pi is the projection of D(x( 17π

90 ), y( 101π 360 ), z( π 3 )) onto Πxy.

• Liping Yuan

lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 53 / 67

Nieuwland constants of Archimedean Solids20

Archimedean Solid Nieuwland Estimate Cuboctahedron C ν(C) > 1.01461 Truncated octahedron O ν(O) > 1.00815 Truncated cube T ν(T ) > 1.02036 Rhombicuboctahedron R ν(R) > 1.00609 Icosidodecahedron I ν(I) > 1.00015 Truncated cuboctahedron U ν(U) > 1.00370 Truncated icosahedron J ν(J ) > 1.00004 Truncated dodecahedron D ν(D) > 1.00014

20Chai Y, Yuan L, Zamfirescu T. Rupert property of Archimedean solids[J].

• Amer. Math. Monthly, 2018, 125(6): 497-504.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 54 / 67

What’s more?

The truncated tetrahedron is Rupert.21

21A paper under review. 22Wacharin Wichiramala, private communications.

• 23G. Huber, K. P. Shultz, J. E. Wetzel, The n-cube is Rupert, Amer. Math.

Monthly, 2018, 125(6), 505-512.

• 24D. Shultz, Largest m-cube in an n-cube, preprint.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 55 / 67

What’s more?

The truncated tetrahedron is Rupert.21 Archimedean dual solids (Catalan solids), Johnson solids.22

21A paper under review. 22Wacharin Wichiramala, private communications.

• 23G. Huber, K. P. Shultz, J. E. Wetzel, The n-cube is Rupert, Amer. Math.

Monthly, 2018, 125(6), 505-512.

• 24D. Shultz, Largest m-cube in an n-cube, preprint.

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What’s more?

The truncated tetrahedron is Rupert.21 Archimedean dual solids (Catalan solids), Johnson solids.22 The n-cube is Rupert.23

21A paper under review. 22Wacharin Wichiramala, private communications.

• 23G. Huber, K. P. Shultz, J. E. Wetzel, The n-cube is Rupert, Amer. Math.

Monthly, 2018, 125(6), 505-512.

• 24D. Shultz, Largest m-cube in an n-cube, preprint.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 55 / 67

What’s more?

The truncated tetrahedron is Rupert.21 Archimedean dual solids (Catalan solids), Johnson solids.22 The n-cube is Rupert.23 Largest m-cube in an n-cube.24

21A paper under review. 22Wacharin Wichiramala, private communications.

• 23G. Huber, K. P. Shultz, J. E. Wetzel, The n-cube is Rupert, Amer. Math.

Monthly, 2018, 125(6), 505-512.

• 24D. Shultz, Largest m-cube in an n-cube, preprint.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 55 / 67

What’s more?

The truncated tetrahedron is Rupert.21 Archimedean dual solids (Catalan solids), Johnson solids.22 The n-cube is Rupert.23 Largest m-cube in an n-cube.24

• Conjecture. Every convex polytope is Rupert.

21A paper under review. 22Wacharin Wichiramala, private communications.

• 23G. Huber, K. P. Shultz, J. E. Wetzel, The n-cube is Rupert, Amer. Math.

Monthly, 2018, 125(6), 505-512.

• 24D. Shultz, Largest m-cube in an n-cube, preprint.

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Unsolved cases for Archimedean solids

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Unsolved cases for Archimedean dual solids

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Unsolved cases for Johnson solids

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Unsolved cases for Johnson solids

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Unsolved cases for Johnson solids

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Unsolved cases for Johnson solids

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Unsolved cases for Johnson solids

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Unsolved cases for Johnson solids

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Unsolved cases for Johnson solids

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Rupert Property of convex bodies

Let K ⊂ I R3 be a convex body, Π ⊂ I R3 a plane, and π : K → Π the

• rthogonal projection onto Π.

The set π−1πbd K is called shadow boundary corresponding to the projection plane Π.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 65 / 67

Rupert Property of convex bodies

Let K ⊂ I R3 be a convex body, Π ⊂ I R3 a plane, and π : K → Π the

• rthogonal projection onto Π.

The set π−1πbd K is called shadow boundary corresponding to the projection plane Π. The shadow boundary S associated with π and Π is normal, if S is a plane parallel to Π, where S is the affine hull of S.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 65 / 67

Rupert Property of convex bodies

Let K ⊂ I R3 be a convex body, Π ⊂ I R3 a plane, and π : K → Π the

• rthogonal projection onto Π.

The set π−1πbd K is called shadow boundary corresponding to the projection plane Π. The shadow boundary S associated with π and Π is normal, if S is a plane parallel to Π, where S is the affine hull of S. The shadow boundary S associated with π and Π will be called balanced if, for the plane Ξ ⊃ xy parallel to Π and determining two closed half-spaces Ξ+ and Ξ−, and for the plane Φ ⊃ xy orthogonal to Π and determining two closed half-spaces Φ+ and Φ−, we have S ∩ Φ+ \ xy ⊂ Ξ+ \ Φ and S ∩ Φ− \ xy ⊂ Ξ− \ Φ.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 65 / 67

Rupert Property of convex bodies

Theorem 1. Any polytope possessing a normal shadow boundary has the Rupert property.

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Rupert Property of convex bodies

Theorem 1. Any polytope possessing a normal shadow boundary has the Rupert property. Any normal shadow boundary is balanced, but the converse is not

• true. Thus, the next result strengthens Theorem 1.

Theorem 2. Any polytope possessing a balanced shadow boundary has the Rupert property.

Liping Yuan lpyuan@hebtu.edu.cn () Selfishness and Rupert Property of convex bodies 2019.05 66 / 67

Rupert Property of convex bodies

Theorem 1. Any polytope possessing a normal shadow boundary has the Rupert property. Any normal shadow boundary is balanced, but the converse is not

• true. Thus, the next result strengthens Theorem 1.

Theorem 2. Any polytope possessing a balanced shadow boundary has the Rupert property. Theorem 3. All convex bodies possessing a balanced shadow boundary, except for those in a nowhere dense subset, have the Rupert property.

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Thanks for your attention!

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