Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?
Victor Alexandrov
math.MG 52B10, 51M20




Why there is no an existence theorem

       for a convex polytope with prescribed

       directions and perimeters of the faces?
    Victor Alexandrov
    Originally submitted: July 25, 2017. Corrected: November 25, 2017
==================================================================================================================================


We choose some special unit vectors 
n_1,…,n_5
in ℝ^3 and denote by ℒ⊂ℝ^5 
the set of all points (L_1,…,L_5)∈ℝ^5 
with the following property:
there exists a compact convex polytope P⊂ℝ^3 
such that the vectors n_1,…,n_5 
(and no other vector) are unit outward normals to the faces of 
P and the perimeter of the face with the outward normal 
n_k is equal to L_k for all k=1,…,5.
Our main result reads that ℒ is not 
a locally-analytic set, i. e., we prove that, for some point 
(L_1,…,L_5)∈ℒ, 
it is not possible to find a neighborhood U⊂ℝ^5 
and an analytic set A⊂ℝ^5 such that 
ℒ∩ U=A∩ U. 
We interpret this result as an obstacle for finding an 
existence theorem for a compact convex polytope with prescribed 
directions and perimeters of the faces.
Mathematics Subject Classification (2010): 52B10; 51M20.
Key words:  Euclidean space, convex polyhedron, perimeter of
a face, analytic set.


1. Introduction and the statement of the main result.
In 1897, Hermann Minkowski proved the following uniqueness 
theorem:
 
Theorem 1 (H. Minkowski, <cit.> and <cit.>).
A convex polytope is uniquely determined, up to
translations, by the directions and the areas of its faces.

Here and below a convex polytope is the convex hull of a finite 
number of points.
By  the direction of a face, we mean the direction of the 
outward normal to the face. 

Theorem 1 has numerous applications and generalizations. 
In order to discuss some of them, we will use the following 
notation.

Let P be a compact convex polytope in ℝ^3 and
n∈ℝ^3 be a unit vector. 
By P^n we denote the intersection of P
and its support plane with the outward normal n. 
Note that P^n is either a vertex, or an edge, 
or a face of P. Accordingly, we say that P^n 
has dimension 0, 1, or 2.

In 1937, A.D. Alexandrov proved several generalizations of 
the above 
uniqueness theorem of Minkowski, including the following

Theorem 2 (A.D. Alexandrov, <cit.> and <cit.>). 
Let P_1 and P_2 be convex polytopes in ℝ^3.
Then one of the following mutually exclusive possibilities realizes:

(i) P_1 is obtained from P_2 by a parallel translation;  

(ii) there exist k=1,2 and a unit vector 
n∈ℝ^3 such that
P_k^n has dimension 2 and, for some 
translation T:ℝ^3→ℝ^3, the formula 
T(P_j^n)⊊ P_k^n 
holds true, where j∈{1,2}{k}.

Note that the formula 
T(P_j^n)⊊ P_k^n 
means that the face P_j^n can be 
embedded inside the face P_k^n by translation 
T as a proper subset.

For more details about Theorems 1 and 2, the reader is referred to <cit.>.

For us, it is important that Theorem 1 is a special case of Theorem 2.
In fact, suppose the conditions of Theorem 1 are fulfilled. 
Then, using the notation of Theorem 2, we observe that, 
for every unit vector n such that P_k^n 
has dimension 2, P_j^n also has dimension 2 
and its area is equal to the area of P_k^n. 
Hence, there is no translation T such that
T(P_j^n)⊊ P_k^n. 
This means that the possibility (ii) in Theorem 2
is not realized. Therefore, the possibility (i) in Theorem 2
is realized and Theorem 1 is a consequence of Theorem 2.

In fact, Theorem 2 has many other consequences, including the following 

Theorem 3 (A.D. Alexandrov, <cit.>).
A convex polytope in ℝ^3 is uniquely 
determined, up to
translations, by the directions and the perimeters of its faces.

For the sake of completeness, we mention that a direct analog 
of Theorem 1 is valid
in ℝ^d for all d⩾ 4; a direct analog 
of Theorem 2 is not valid
in ℝ^d for every d⩾ 4; in ℝ^3, 
a refinement of Theorem 2 
was found by G.Yu. Panina <cit.> in 2008.

We explained above that uniqueness Theorems 1 and 3 are similar 
to each other and both follow from Theorem 2. 
In the rest part of this section  we explain the difference 
that appears when we are interested in existence results 
corresponding to uniqueness Theorems 1 and 3.

In 1897, Hermann Minkowski also proved the following existence 
theorem:
 
Theorem 4 (H. Minkowski, <cit.> and <cit.>).
Let unit vectors 
n_1,…,n_m in ℝ^3
and real numbers F_1,…,F_m satisfy the following conditions: 

(i) n_1,…,n_m 
are not coplanar and no two of them coincide with each other;

(ii) F_k is positive for every k=1,…,m;

(iii) ∑_k=1^m F_k n_k=0.

Then there exists a convex polytope 
P⊂ℝ^3 such that 
n_1,…,n_m (and no other vector)
are outward face normals for P and F_k is the area of 
the face with
outward normal n_k for every k=1,…,m.

For the sake of completeness, we mention that a direct 
analog of Theorem 4 is valid
in ℝ^d for all d⩾ 4. 

For more details about Theorem 4, the reader is referred 
to <cit.>.

Recall that a set A⊂ℝ^d is said to be algebraic 
if A={x∈ℝ^d: p(x)=0} for some polynomial
p:ℝ^d→ℝ, 
and A is said to be locally-algebraic if, for every x∈ A, 
there is a neighborhood
U⊂ℝ^d and an algebraic set 
A_0⊂ℝ^d such that A∩ U=A_0∩ U.
 
For a given set {n_1,…,n_m} 
of vectors satisfying the condition (i) of Theorem 4, 
denote by ℱ(n_1,…,n_m)
the set of all points (F_1,…,F_m)∈ℝ^m 
such that there exists a convex polytope P⊂ℝ^3
for which n_1,…,n_m (and no 
other vector) are the outward face normals for P, and F_k 
is the area of the face with the outward normal 
n_k for every k=1,…,m.
The set ℱ(n_1,…,n_m)⊂ℝ^m
can be referred to as a natural configuration space of convex 
polytopes (treated up to translations) with prescribed set 
{n_1,…,n_m} 
of outward unit normals when a polytope is determined by
the areas F_1,…,F_m of its faces.

From Theorem 4, it follows immediately that 
the set ℱ(n_1,…,n_m)⊂ℝ^m
is locally-algebraic for 
every set {n_1,…,n_m} 
of vectors satisfying the condition (i) of Theorem 4.
In fact, we can define the algebraic set A_0 as the zero set
of the quadratic polynomial 

    ∑_j=1^3(∑_k=1^m 
    (n_k,e_j) F_k)^2,

where (n_k,e_j) stands for the
standard scalar product in ℝ^3 and 
{e_1, e_2, e_3}
is the standard orthonormal basis in ℝ^3.

Recall that a set B⊂ℝ^d is said to be analytic 
if B={x∈ℝ^d: φ(x)=0} for some 
real-analytic function φ:ℝ^d→ℝ, 
and B is said to be locally-analytic if, for every x∈ B, 
there is a neighborhood U⊂ℝ^d and an analytic 
set B_0⊂ℝ^d such that B∩ U=B_0∩ U.

Obviously, every locally-algebraic set is locally analytic.

Let unit vectors n_1,…,n_5
in ℝ^3 be defined by the formulas

    n_1=(0,0,-1),  n_2=   (1/√(2),0,1/√(2)),  n_3=(-1/√(2),0,1/√(2)),
    n_4=   (0,1/√(2),1/√(2)),  n_5=(0,-1/√(2),1/√(2)).

For convenience of the reader, the vectors n_1,…,n_5
are shown schematically in Figure 1.

By ℒ=ℒ(n_1,…,n_5)⊂ℝ^5 
we denote the set of all points (L_1,…,L_5)∈ℝ^5 
with the following property:
there exists a convex polytope P⊂ℝ^3 such that 
the vectors n_1,…,n_5 (and no 
other vector) are the unit outward normals to the faces of P, 
and L_k is the perimeter of the face with the outward normal 
n_k for every k=1,…,5.
The set ℒ(n_1,…,n_5)⊂ℝ^5
can be referred to as a natural configuration space of convex 
polytopes (treated up to translations) with prescribed set 
{n_1,…,n_5} 
of outward unit normals, when a polytope is determined by
the perimeters L_1,…,L_5 of its faces.

The main result of this article reads as follows:

Theorem 5. 
Let the vectors n_1,…,n_5 
be given by the formulas (1). Then the set 
ℒ(n_1,…,n_5)⊂ℝ^5 
is not locally-analytic.

From our point of view, Theorem 5 explains why a general 
existence theorem is not known which 
determines a convex polytope in ℝ^3 via unit 
normals and perimeters of its faces. 
The reason is that no analytic condition, similar to 
the condition (iii) in Theorem 4, does exist.

2. Auxiliary constructions and preliminary results.
Let P⊂ℝ^3 be a convex polytope such that 
the vectors n_1,…,n_5 
defined by the formulas (1) (and no other vector) are the 
unit outward normals to the faces of P.
For k=1,…,5, denote by π_k the 2-dimensional
plane in ℝ^3 containing the face of P with the 
outward normal n_k.

The straight lines π_1∩π_2 and π_1∩π_3
are parallel to the vector e_2=(0,1,0),
and the straight lines π_1∩π_4 and π_1∩π_5
are parallel to the vector e_1=(1,0,0).
Hence, the face P∩π_1 is a rectangle.
Computing the angles between the vectors n_1
and n_k for k=2,…, 5, we conclude that
the dihedral angle attached to any edge of the face P∩π_1 
is equal to π/4.
Now it is clear that the polytope P can be of one of the three
types schematically shown in Figure 2. 


In order to be more specific, we put by definition
A=π_1∩π_2∩π_5,
B=π_1∩π_2∩π_4,
C=π_1∩π_3∩π_4, and
D=π_1∩π_3∩π_5.
Denote by 2x the length of the straight line segment AB,
and by 2y the length of the straight line segment BC.
We say that the polytope P is of Type I, if x<y;
is of Type II, if x=y; and is of Type III, if x>y.
Polytopes P of Types I–III are shown schematically  in Figure 2. 

Denote by 
ℒ_I(n_1,…,n_5) 
(respectively, by 
ℒ_II(n_1,…,n_5)
and ℒ_III(n_1,…,n_5))
the set of all points (L_1,…,L_5)∈ℝ^5 such that
there exists a convex polytope P⊂ℝ^3 
of Type I (respectively, of Type II or Type III) 
such that the vectors
n_1,…,n_5 
(and no other vector) are outward unit normals
to the faces of P, and L_k is the perimeter
of the face of P with the outward normal n_k
for every k=1,…, 5.
Below, we use also the following notation

    v_I   =   (2, -(3+2√(3)), -(3+2√(3)), 5, 5),
    v_II   =   (2(√(3)-1), 1,1,1,1),
    v_III   =   (2,5,5,-(3+2√(3)), -(3+2√(3))).

Note that in this paper the notation AB can as well denote
the straight line segment and its length.

Lemma 1. 
Let the set {n_1,…,n_5} 
of unit vectors in ℝ^3 be defined by the formulas (1).
Then the following three statements are equivalent to each other:

(i) (L_1,…,L_5)∈ℒ_I(n_1,…,n_5);

(ii) L_1=(2√(3)-3)L_2+L_4, L_2=L_3, L_4=L_5, L_4>L_2>0;

(iii) (L_1,…,L_5)=αv_I +βv_II
for some α, β∈ℝ 
such that β>(3+2√(3))α>0.

Proof :
Suppose the statement (i) of Lemma 1 holds true.
In addition to the notation introduced above in Section 2, let 
E=π_2∩π_4∩π_5, F=π_3∩π_4∩π_5,
G be the base of the perpendicular dropped from E on the 
edge AB, 
H be the base of the perpendicular dropped from E on the face
ABCD, 
and K be the base of the perpendicular dropped from E on 
the edge BC, see Figure 3. 

Using this notation, we obtain easily 

    BG   =   GH=EH=BK=x,
    
    BH   =   x√(2),
    
    BE   =   x√(3),
    
    EF   =   BC-2 BK=2y-2x,
    
    L_1   =   2AB+2BC=4x+4y,
    
    L_2   =   L_3=AB+2BE=2(1+√(3))x,
    
    L_4   =   L_5=BC+EF+2BE=2(√(3)-1)x+4y.

Eliminating x and y from the last three formulas, we get
L_1=(2√(3)-3)L_2+L_4.
Since the relations L_2=L_3, L_4=L_5, and L_4>L_2>0
are obvious for every polytope P of Type I,
we conclude that the statement (i) implies the statement (ii).

Now suppose the statement (ii) of Lemma 1 holds true.
We, first, find the numbers x and y that satisfy the 
relations (2)–(4) and, second, construct a convex 
polyhedron P of Type I for which L_1,…, L_5 are 
the perimeters of the faces.

In accordance with (3), we put by definition 
x=L_2(√(3)-1)/4. Since L_2>0, x>0.
In accordance with (4), we put by definition
y=L_2(√(3)-2)/4+L_4/4.
Since L_1=(2√(3)-3)L_2+L_4, the condition (2) 
is satisfied: 4x+4y=(2√(3)-3)L_2+L_4=L_1.
Using the inequalities L_4>L_2>0, we obtain
4y=L_4+(√(3)-2)L_2>L_2+(√(3)-2)L_2=(√(3)-1)L_2>0.
Hence, there exists a rectangle ABCD in ℝ^3 
such that the edge AB is parallel to the vector 
e_2=(0,1,0) and its length is equal to 2x,
and the edge BC is parallel to the vector 
e_1=(1,0,0) and its length is equal to 2y.

Denote by π_1 the plane that contains ABCD. 
Obviously, π_1 is perpendicular to n_1.
Denote by π_2 the plane perpendicular to n_2
and containing AB.
Denote by π_3 the plane perpendicular to n_3
and containing CD.
Denote by π_4 the plane perpendicular to n_4
and containing BC.
And denote by π_5 the plane perpendicular to n_5
and containing AD.
The five planes π_1,…,π_5 determine a compact 
convex polyhedron with outward normals 
n_1,…,n_5.
Denote it by P. 
According to the statement (ii), L_4>L_2.
Hence, 4y=L_4+(√(3)-2)L_2>L_2+(√(3)-2)L_2=(√(3)-1)L_2=4x.
Thus, P∈ℒ_I and
the statement (ii) implies the statement (i).

So, we have proved that the statements (i) and (ii)
are equivalent to each other.

In order to prove that the statements (ii) and (iii)
are equivalent to each other, we observe that the equations 
L_1=(2√(3)-3)L_2+L_4, L_2=L_3, and L_4=L_5
from the statement (ii) define a 2-dimensional plane in 
ℝ^5. Denote this plane by λ_I.
Moreover, the vectors v_I
and v_II constitute an orthogonal basis 
in λ_I.
This means that every vector (L_1,…,L_5)∈λ_I
can be uniquely written in the form 
αv_I+βv_II.
Direct calculations show that the inequalities
L_4>L_2>0 from the statement (ii) are equivalent 
to the inequalities β>(3+2√(3))α>0
from the statement (iii). □

Lemma 2. 
Let the set {n_1,…,n_5} 
of unit vectors in ℝ^3 be defined by the formulas (1).
Then the following three statements are equivalent to each other:

(i) (L_1,…,L_5)∈ℒ_II(n_1,…,n_5);

(ii) L_1=2(√(3)-1)L_2, L_2=L_3=L_4=L_5>0;

(iii) (L_1,…,L_5)=γv_II
for some γ∈ℝ 
such that γ>0.

Proof is left to the reader.
It can be obtained by arguments similar to those used above
in the proof of Lemma 1. 
But in fact, it is sufficient to observe that Lemma 2
is the limit case of Lemma 1 as L_4 approaches L_2.

Lemma 3. 
Let the set {n_1,…,n_5} 
of unit vectors in ℝ^3 be defined by the formulas (1).
Then the following three statements are equivalent to each other:

(i) (L_1,…,L_5)∈ℒ_III(n_1,…,n_5);

(ii) L_1=L_2+(2√(3)-3)L_4, L_2=L_3, L_4=L_5, L_2>L_4>0;

(iii) (L_1,…,L_5)=δv_III +εv_II
for some δ, ε∈ℝ 
such that ε>(3+2√(3))δ>0.

Proof is left to the reader.
It can be obtained by arguments similar to those used above
in the proof of Lemma 1. 
But in fact, it is sufficient to observe that if we rotate 
a polytope of Type III around the vector 
e_3=(0,0,1) to the angle π/2,
we get a polytope of Type I and can apply Lemma 1 to it. 

In the proof of Lemma 1, we denoted by λ_I 
the 2-dimensional subspace in ℝ^5
which is spanned by the vectors 
v_I and v_II.
Now we denote by λ_II 
the 1-dimensional subspace in ℝ^5 
spanned by v_II
and denote by λ_III 
the 2-dimensional subspace spanned by v_II
and v_III. 
  
Lemma 4. 
λ_II=λ_I∩λ_III.

Proof :
Each subspace λ_I, λ_II, and λ_III
contains v_II.
Hence, (λ_I∩λ_III)⩾ 1. 
On the other hand, λ_I=λ_III=2. 
Hence, (λ_I∩λ_III) is equal to either 1 
or 2. 

Suppose (λ_I∩λ_III)=2.
Then λ_I=λ_III.
Hence, the vectors v_I, v_II, 
and v_III are linearly dependant.
But this is not the case because the 3×3 minor 
composed of the first, third and fifth columns of the matrix

    [   v_I;  v_II; v_III ]
    =
    [          2 -(3+2√(3)) -(3+2√(3))          5          5;  2(√(3)-1)          1          1          1          1;          2          5          5 -(3+2√(3)) -(3+2√(3)) ]

is non-zero. Hence, (λ_I∩λ_III)=1, and
λ_II=λ_I∩λ_III. □

3. Half-branches of analytic sets and the proof of Theorem 5. 
Let A be a one-dimensional analytic set, and x∈ A. 
For every sufficiently small open ball U with center x, 
A∩ (U{x}) has a finite number of 
connected components A_1,…,A_k such that x belongs
to the closure of A_j for every j=1,…, k.
These A_j are called the half-branches of A 
centered at x.
It is known that the number of half-branches 
of a one-dimensional analytic set centered at a point is even, 
see, e. g. <cit.>.

For completeness, we mention that a comprehensive exposition of a similar result for 
algebraic sets of dimension 1 may be found in <cit.>.

Proof of Theorem 5 :
Let Λ be the straight line in ℝ^5
defined by the formula
Λ={x∈ℝ^5| x=
v_II+tv_I  for some t∈ℝ}.

Our proof is by contradiction.
Suppose the set 
ℒ(n_1,…,n_5)⊂ℝ^5 
is locally-analytic.
Then  
Λ∩ℒ(n_1,…,n_5)
is also locally-analytic.
Moreover, it is one-dimensional, contains the point
v_II, and has only one half-branch centered at
v_II. 

Let us explain the last statements in more details.
From the definition of polytopes of Types I–III we know that

    ℒ(n_1,…,n_5)=
    ℒ_I(n_1,…,n_5)∪ℒ_II(n_1,…,n_5)∪ℒ_III(n_1,…,n_5).

From Lemma 1 we know that 
ℒ_I(n_1,…,n_5)
is an angle on the 2-dimensional plane λ_I⊂ℝ^5.
From Lemma 3 we know that 
ℒ_III(n_1,…,n_5)
is an angle on the 2-dimensional plane λ_III⊂ℝ^5.
These angles are glued together along the ray
ℒ_II(n_1,…,n_5) (see Lemma 2), and
no 2-dimensional plane contains the both of them (see Lemma 4). 
The line Λ lies in the plane λ_I and passes through the point
v_II. Hence, for every sufficiently small open ball 
U⊂ℝ^5 with center v_II, 

    U∩Λ∩ℒ(n_1,…,n_5)=
    {v_II}∪(U∩Λ∩ℒ_I(n_1,…,n_5)).

This formula means that we may obtain
U∩Λ∩ℒ(n_1,…,n_5) in the following way:
first, we divide the straight line Λ into two rays by the point
v_II; then we observe that only one of these rays has at least one
common point with the angle ℒ_I(n_1,…,n_5) 
and select that ray; at last, we intersect the ray selected with U. 

From this description, it is clear that 
U∩Λ∩ℒ(n_1,…,n_5)
is the half-branch of the locally-analytic set 
ℒ(n_1,…,n_5)⊂ℝ^5
centered at v_II. 
Moreover, this is the only half-branch centered at v_II. 
This contradicts to the fact that the number of half-branches 
of a locally-analytic set centered at a point is even, see <cit.>.
□

Remark :
The proof of Theorem 5 provides us with a new, more technical, answer to the 
question of the title of this article.
As a part of the proof of Theorem 5, we demonstrated that the set 
ℒ(n_1,…,n_5)⊂ℝ^5 is not convex.
In Section 1, we mentioned that ℒ(n_1,…,n_5)
can be considered as a natural configuration space of convex 
polytopes (treated up to translations) with prescribed 
outward unit normals and perimeters of its faces.
The reader, familiar with the proof of Theorem 4 given in <cit.>, 
may remember that convexity of the analogous `natural configuration space'
ℱ(n_1,…,n_m)⊂ℝ^m
plays an important role in that proof.

8

Al37Alexandrov, A.D.:
An elementary proof of the Minkowski and some other theorems on convex polyhedra (in Russian).
Izv. Akad. Nauk SSSR, Ser. Mat. No. 4, 597–606 (1937). 
JFM 63.1234.02

Al96Alexandrov, A.D.:
Selected works. Part 1: Selected scientific papers.
Gordon and Breach Publishers, Amsterdam (1996).
MR1629804, Zbl 0960.01035

Al05Alexandrov, A.D.:
Convex polyhedra. 
Springer, Berlin (2005).
MR2127379, Zbl 1067.52011

BCR8Bochnak, J.; Coste, M.; Roy, M.-F.:
Real algebraic geometry.
Springer, Berlin (1998).
MR1659509, Zbl 0912.14023

Mi97Minkowski, H.:
Allgemeine Lehrsätze über die convexen Polyeder. 
Gött. Nachr. 198–219 (1897).
JFM 28.0427.01

Mi11Minkowski, H.:
Gesammelte Abhandlungen von Hermann Minkowski. Band I.
Teubner, Leipzig (1911).
JFM 42.0023.03

Pa08Panina, G.:
A.D. Alexandrov's uniqueness theorem for convex polytopes and its refinements.
Beitr. Algebra Geom. 49, No. 1, 59–70 (2008).
MR2410564, Zbl 1145.52007

Su71Sullivan, D.:
Combinatorial invariants of analytic spaces. 
Proc. Liverpool Singularities-Sympos. I, 
Dept. Pure Math. Univ. Liverpool 1969–1970, 165–168 (1971).
MR0339241, Zbl 0227.32005




Victor Alexandrov

Sobolev Institute of Mathematics

Koptyug ave., 4

Novosibirsk, 630090, Russia

and

Department of Physics

Novosibirsk State University

Pirogov str., 2

Novosibirsk, 630090, Russia

e-mail: alex@math.nsc.ru

