BERKELEY MATH CIRCLE
GEOMETRIC COMBINATORICS
Tatiana Shubin
shubin@math.sjsu.edu
Geometric Combinatorics is a growing body of mathematics concerned with
counting properties of geometric objects described by a finite set of building blocks.
Examples include bounded polyhedra and the convex hulls of finite sets of points and
complexes built up from them. Other examples include arrangements and intersections of
convex sets and other geometric objects. Geometric combinatorics has many connections
to linear algebra, discrete mathematics, mathematical analysis, and topology, and it has
applications to economics, game theory, and biology. Many of the problems encountered
within geometric combinatorics are easy to state yet there are lots of problems that are
extremely hard to solve, including a great many that remain open despite the efforts of
some leading mathematicians. The following definitions, theorems, and problems should
give you some flavor of this branch of mathematics.
A convex planar figure is the intersection of a number (finite or infinite) of halfplanes. The intersection of a finite number of half-planes is a convex polygon. Convex 3d figures and convex polyhedra are defined similarly (replacing half-planes with halfspaces). Equivalently, a figure F is convex if for every two points A and B of F, F
contains the entire line segment AB.
If S is any set of points, the convex hull of S (denoted by conv(S)) is the smallest
convex figure containing S (clearly, conv(S) is the intersection of all convex sets
containing S).
n
A convex combination of points a1, , a2 , ... , an is a linear combination
 a
i 1
i i
in
which the coefficients i are non-negative and sum to 1.
THEOREM (Carathéodory’s Theorem). For S in
convex combination of at most n + 1 points of S.
n
, each point of conv(S) is a
THEOREM (Radon’s Lemma). Let S be a set of size n + 2 or greater in
Then S can be partitioned in two sets R and B (red and blue) such that
conv( R)  conv( B)   .
n
.
THEOREM (Helly’s Theorem). Suppose S = { S1 , S2 , ..., Sm } is a set of convex
sets in n , such that every subset of S of size n + 1 has non-empty intersection. Then
m
S
i
is non-empty.
i 1
1
Let P be a polytope, and let f k  f k (P) denote the number of k-faces of P. (Thus
f 0 is the number of vertices, f1 the number of edges, etc.)
THEOREM (Euler-Poincaré Formula). For any n-polytope P,
f 0  f1  ...  (1) n 1 f n 1  1  (1) n .
THEOREM (Cauchy’s Uniqueness Theorem). Two convex polyhedra of the
same combinatorial type (i.e., having congruent faces combined in the same order) are
congruent to each other.
THEOREM (Sabitov’s Theorem). Let P be a polyhedron with triangular faces.
There exists a polynomial FP ( x)  x n  a1 x n 1  a2 x n  2  ...  an , with the coefficients
ai , i  1, 2, ... , n that can be found from the length of the edges of P using addition,
subtraction, multiplication and division only, such that the volume of P is one of the roots
of FP (x) .
COROLLARY: Polyhedra of the same combinatorial type have the same
volume.
Let be a set of polygons S1 , S2 , ... , Sn such that every edge of each polygon Si
is identified with exactly one other edge of a polygon S j of (i might be the same as j).
Furthermore, this identification of the edges must satisfy the following two conditions:
(1) the identified edges have the same length;
(2) for any 1  i, j  n it is possible to get from Si to S j going through polygons that
have identified edges.
Such a collection is called a net.
THEOREM (Cauchy-Aleksandrov’s Theorem) Two convex polyhedra with the
same net are congruent.
THEOREM (Aleksandrov’s Net Theorem) Suppose that a net satisfies the
following two conditions.
(1) V – E + F = 2 (here F is the number of polygons in the net; E is the number of
edges where two identified edges are counted as 1; V is the number of vertices,
where all identified vertices are counted as 1);
(2) The sum of all angles emanating from the same vertex of is at most 2 .
Then there exists a convex polyhedron corresponding to the net .
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PROBLEMS
1.
Prove that if each three of n points in a plane can be enclosed in a circle
of radius 1, then all n points can be enclosed in such a circle.
2.
Let A, B be points in an annulus bounded by concentric circles with radii
r  R , and suppose that AB = 1. Find the smallest possible value of (AOB ) .
3.
(a) Given n  4 points in a plane. Prove that we can choose 3 out of these
points so that they form a triangle with one angle at most   45 . Also prove that
there exists such a configuration of 4 points in a plane that every 3 of them form a
triangle with each angle at least 45 .
(b) Same as in part (a), except that n  5 and   36 .
(c) Same as in part (a), except that n  6 and   30 .
(d) Prove that for any n points in a plane ( n  3 ), it is possible to choose 3 points
180
so that they form a triangle such that one of its angles is no larger than
. Prove
n
also that there exists such a configuration of n points that in any triangle formed by 3
180
of these points each angle is at least
.
n
4. Let M1 , M 2 , ... , M n be points. Find all possible values of n such that all these
points can be positioned in such a way that none of the angles M i M j M k is
obtuse.
(a) assume that all these points lie in a plane;
(b) assume that the points can be placed anywhere in 3-d space.
5. Same as problem 3, except that now all the angles M i M j M k must be acute.
6. Same as problem 3, except that now we require all the triangles M i M j M k be
right triangles.
7. Same as problem 3, but all the triangles M i M j M k are required to be obtuse.
8. Prove that
a. any convex polygon of area 1 can be covered by a parallelogram of area 2;
b. a triangle of area 1 cannot be covered by a parallelogram of area less than
2.
3
9. Let M be a convex n-gon with vertices A1 , A2 , ... , An . Let Ai be the midpoint
of Ai Ai 1 , (i  1, 2, ... , n; assume i  1  1 when i  n) , and let M  be the
polygon with vertices Ai, i = 1, 2, …, n. Let P and S ( P  and S  ) denote
the perimeter and area of M ( M  ), respectively. Prove that
1
(a) P  P  P for any n  3 ;
2
1
(b) S  S   S for any n  4 .
2
Is it possible to improve these inequalities?
10. Let M be a convex polygon with perimeter P and area S. Prove that there
S
exists a circle of radius
with all points in M.
P
11. .
(a) Prove that of all triangles of a given perimeter P, an equilateral
triangle has the largest area.
(b) Prove that of all the convex quadrilaterals of a given perimeter P, a square
has the largest area.
(c) Prove that of all the convex n-gons of a given perimeter P, a regular n-gon
has the largest area.
12. Let k be a unit circle. Find the largest number n of unit circles k1, k2, … , kn
such that ki touches k for every i= 1, 2, … , n, and
a. ki and kj do not intersect for any i  j , 1  i, j  n ;
b. the interior of ki does not contain the center of kj for any
i  j , 1  i, j  n .
13. Let s be a square of side 1. Find the largest number n of squares s1, s2, … , sn
such that si is a square of side 1 and it touches s for every i= 1, 2, … , n, and
si, sj do not intersect for any i  j , 1  i, j  n .
14. Let F be a planar figure, and let h(F) denote the maximal number of copies
of F obtained from F by translation and such that every copy touches F and
no pair of these copies intersect each other.
(a) Find h(T) where T is a triangle.
(b) Find h(S) where S is a square.
15. Let k be a circle of radius 2R. Find the least number n of circles k1, k2, … ,
kn of radius R each so that k can be covered by k1, k2, … , kn.
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16. Let k be a unit circle, and let k1, k2, … be circles of radii less than 1. Let n
be the minimal number such that k1, k2, … , kn cover k. Prove that
(a) n  3 ;
3
(b) if the radii of k1, k2, … are less than
, then n  4 ;
2
2
(c) if all the radii are less than
, then n  5 .
2
Is it possible to improve the estimates in parts (a), (b), and (c)?
17. Let k be a unit circle, and let k1, k2, … be circles of radii r1, r2, … . Prove that
1
(i  1,2,...) , then it is not possible to place 2 of these circles ki,
a. If ri 
2
kj inside k in such a way that ki and kj do not intersect.
b. If ri  2 3  3 , then it is not possible to place 3 circles ki , k j , km inside k
in such a way that no two of ki , k j , km intersect each other.
18. Let M be a convex polygon and suppose that M is not a parallelogram. Prove
that M can be placed inside a triangle obtained by producing 3 of its sides.
19. Suppose that there are n points in a plane such that each two are less than 1
3
unit apart. Prove that all these points can be enclosed in a circle of radius
.
3
20. Given n parallel line segments in the same plane, prove that if there exists a
line that intersects each three of them, then there is a line intersecting all these
line segments.
21.
(a) Find Rn , the radius of the smallest circle k such that it is possible to
place n points inside k (or on k) in such a way that one of the points is in
the center of k, and the distance between any two of these points is at least 1.
(b) How many points can be placed inside (or on) a circle of radius 2 in
such a way that one of these points is in the center of the circle, and the
distance between any two of the points is at least 1?
22. Suppose there is a non-transparent spherical planet P whose diameter is d. Is it
possible to place 8 observation stations on the surface of the planet in such a
way that any object approaching the planet would be seen by at least 2 of these
stations when the object is a distance d above the planet’s surface?
23. Prove that the minimal number of vertices for a polyhedron of a torus type is 7.
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24. An art gallery is in a polygonal room (with finitely many sides, and possibly
non-convex) and there is one painting hung on each wall. Suppose that for any 3
paintings, there is a point in the gallery where those 3 paintings are visible. Is
there always a point of the gallery from which all paintings are visible?
25. Can a convex bounded polyhedron have 7 edges?
26. Take an n-gon and triangulate it (i.e., subdivide it into triangles) so that the
triangle corners are tangent to the vertices of the n-gon. How many triangles do
you need? And in how many different ways can this be done? (Find a formula in
terms of n.)
27.
Suppose you try to subdivide a cube into tetrahedra whose corners are all
tangent to corners of the cube. What is the minimum and maximum number of
tetrahedra that will accomplish this? Prove your claims.
SOME OPEN PROBLEMS
I.
Throw k points down in the unit square and find the area of the largest
convex set in the square containing none of the k points. Let f(k) be the
minimum (of the largest areas) over all sets of k points. Find good upper
1
2
and lower bounds on f(k). (For k = 3 it is known that  f (3) 
.)
3
4
II.
What is the largest area that an n-gon of unit diameter can have?
III.
Can every set of 8 points in the plane be partitioned to form 2 triangles
and a line segment so that the segment cuts the interior of both triangles?
(Notice: this is a Radon relative).
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SOME ANSWERS
1
1
, then min   2 sin 1
;
R
2R
1
R2  r 2  1
If R  1  r  R  , then min   cos 1
;
R
2 Rr
if r  R  1 , then min   0 .
2.
If r  R 
4.
(a) n = 3, 4;
5.
(a) n = 3;
6.
(a) n = 3, 4;
7.
n can be any natural number  3 .
12.
(a) 6;
13.
8.
14.
(a) 6;
15.
7.
21.
(a) Rn  1 for n  2, 3, ... , 7 ; Rn 
(b) n  8 .
(b) n = 3, 4, 5.
(b)
n = 4.
(b) 12.
(b) 8.
1

csc
for n  8,9,10,11 .
2
n 1
(b) 19.
22.
Yes
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