Simple polytopes

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Polytopes & f-vectors The Steiner point Generalized permutahedra Tree Posets

On the Geometric Combinatorics of Steiner Points and a Conjecture of Gr¨ unbaum

Work in progress with Raman Sanyal

Aenne Benjes

May 20th 2021

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Polytopes & f-vectors

The Steiner point

Generalized permutahedra

Tree Posets

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Polytopes & f-vectors

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Polytopes

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The f-vector of a polytope

Definition

For ad-polytope P thef-vectorf(P) = (f0,f1, . . . ,fd), is defined by

f_i =|{F ⊆P face : dim(F) =i}|.

Theorem (Euler–Poincar´e relation) The relation

f0−f1+· · ·+ (−1)^dfd =fd

is the unique linear relation fulfilled by all f -vectors of d -polytopes.

Euler’s polyhedron formula: f₀−f₁+f₂= 2

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Simple polytopes

Definition

Ad-polytope P is simpleif every vertex is incident to exactly d edges. For ad-polytopeP theh-vector h(P) = (h0,h1, . . . ,hd) is defined by

d

X

i=0

h_i(t+ 1)ⁱ =

d

X

i=0

f_itⁱ.

Theorem (Dehn–Sommerville relations)

Theb^d+1₂ crelations

hi =hd−i for 0≤i ≤ b^d₂c

give a basis for all linear relations on h-vectors of simple d -polytopes.

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An interpretation of h-vectors

LetP be a simpled-polytope. We call c ∈R^d generic, if c^tu 6=c^tv for all u,v ∈V(P).

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An interpretation of the h-vector

For a vertexv ∈V(P) letF(v,c) =F ⊆P be the inclusion-maximal face, such that

F^c :={x ∈F :c^tx ≥c^ty for all y∈F}={v}.

Proposition (McMullen)

If P is a simple d -polytope and c∈R^d generic, then hi =|{(F,v) : dim(F) =i and F =F(v,c)}|.

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The Steiner point

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History

• 1840 Steiner introduced the Steiner point (also:

”Kr¨ummungsschwerpunkt”) within an extremal problem for plane convex regions.

• For a plane convex regionK, he defineds(K) as a the centroid of its boundary.

• Later, Shephard defined it for bounded convex sets in R^d via an integral.

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The Steiner point of a convex set

For a convex, bounded setK ⊆R^d theSteiner pointis defined by s(K) = 1

vol_d(B_d) Z

c∈S^d−1c h(c,K)dω, whereh(c,K) is the support function

h(c,K) = max

x∈K c^tx.

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Uniqueness of the Steiner Point

Theorem (Shephard, Schneider)

Let K,K₁,K₂⊆R^d be convex bounded sets. Then (i) s(K1+K2) =s(K1) +s(K2),

(ii) s(AK) =As(K) for rigid motions A and

(iii) s is continuous with respect to Hausdorff metric.

Moreover, if any map

f :{K ⊆R^d convex, bounded set} →R^d satisfies (i)-(iii), then

f =s.

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The normal cone

Definition

For a vertexv ∈V(P) we define the normal cone ofP atv by N_vP ={c ∈R^d :c^tv ≥c^tx for all x ∈P}.

Definition (External angle)

α(v,P) = vol_d−1(NvP∩S^d−1) vol_d−1(S^d−1) .

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The Steiner Point of a polytope

s(P) = ^X

v∈V(P)

α(v,P)v.

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The Steiner Point of a polytope

Asα(v,P)≥0 and

X

v∈V(P)

α(v,P) = 1,

it follows thats(P)∈P.

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An analogue of the Euler–Poincar´ e relation

LetP be a d-polytope. We define s_i(P) := ^X

F⊆P:

dim(F)=i

s(F) = ^X

v∈V(P)

A(v,i) v,

whereA(v,i) :=^Pv∈F⊆P:

dim(F)=i

α(v,F).

Theorem (Gr¨unbaum) The linear relation

s₀(P)−s₁(P) +· · ·+ (−1)^ds_d(P) =s_d(P)

is the unique linear relation that s0(P),s1(P), . . . ,s_d(P) fulfil for all d -polytopes P.

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A theorem and a conjecture of Gr¨ unbaum

Theorem (Gr¨unbaum)

If P ⊆R^d is a simple d -polytope, then

s_i(P) =

i

X

k=0

(−1)^k d i−k

!

s_k(P) for 0≤i ≤d

and for i= 1,3, . . . ,m the linear relations are linearly independent, where m is the largest odd integer not exceeding d .

Moreover, Gr¨unbaum conjectures that the relations form a basis for all linear relations on Steiner points of simpled-polytopes and thus are an analogue of the Dehn–Sommerville relations.

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This motivated us to definet0(P),t1(P), . . . ,t_d(P)∈R^d by

d

X

i=0

ti(P)(x+ 1)ⁱ =

d

X

i=0

si(P)xⁱ.

so we can restate the relations as

ti(P) =td−i(P) for 0≤i ≤d.

Obviously for 0≤i ≤ b^d₂c they are linearly independent and the remaining question is, if they form a basis for all linear relations on t0(P), . . . ,t_d(P) for simple d-polytopesP.

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t

₀

(P ), t

₁

(P ), . . . , t

_d

(P ) as weighted sums over the vertices

LetP be a simpled-polytope. Similar to si(P) we want to write ti(P) as a weighted sum over the vertices:

ti(P) = ^X

v∈V(P)

B(v,i) v.

IsB(v,i)≥0 and how can the weightsB(v,i) be interpreted?

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The pure normal cone

LetP be a simpled-polytope, v ∈V(P) a vertex andF ⊆P a face ofP with v ∈F.

Definition (Pure normal cone ofv atF)

Nˆ_v(F) :=N_vF \ ^[

F⊆G⊆P

N_vG.

The pure normal cone ofv atF is the half-open cone of those c ∈R^d, such that F =F(v,c).

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The pure external angles

Definition (Pure external angle of v at F)

β(v,F) := vol_d−1( ˆN_vF ∩S^d−1) vol_d−1(S^d−1) .

Proposition (B.,Sanyal)

Let P be a simple d -polytope and v ∈V(P) a vertex. Then B(v,i) = ^X

v∈F⊆P:

dim(F)=i

β(v,F)

and thus B(v,i)≥0.

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A connection between the h-vector and pure external angles

Proposition (B.,Sanyal)

If P is a simple d -polytope, then^P_v∈V_(P)B(v,i) =h_i(P) . Suppose there area0, . . . ,ad,C ∈R, such that for any simple d-polytope P we have^P^d_i=0a_it_i(P) =C.Then for all x ∈R^d

C =

d

X

i=0

aiti(P+x) =

d

X

i=0

ai

X

v∈V(P)

B(v,i)(v+x)

=C +x

d

X

i=0

ai

X

v∈V(P)

B(v,i) =C +x

d

X

i=0

aihi

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Gr¨ unbaums conjecture is true!

Theorem (B., Sanyal)

Theb^d+1₂ crelations

t_i =td−i for 0≤i ≤ b^d₂c

form a basis for all linear relations on t0, . . . ,t_d of simple d -polytopes.