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Which are some (research level) open problems in Euclidean geometry ?

(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)

I should clarify a bit what I mean by "Euclidean geometry". By this term I mean, loosely, the study of the geometry of certain subsets of Euclidean space $\mathbb{E}^n$ from a point of view which is either the "classical" one (i.e. axiomatic), or one that involves more modern tools, but the problem in question has not to be "clearly" a problem within some other branch of maths such as differential or algebraic geometry, algebraic or general topology, analysis, or measure theory.

Some examples to clarify:

  • the study of configurations of lines or affine subspaces is EG; but the algebro-topological study of hyperplane arrangements is not.
  • plane conics as defined via their metric property are objects of EG; but "algebraic curves" are not, unless they're difined by some "elementary enough property" (intentionally vague) involving the Euclidean metric.
  • root systems of Lie algebras are EG.
  • polyhedral cones are EG.
  • polytopes are EG.
  • tassellations of space with polytopes or analogous objects are in EG.
  • minimal surfaces in $\mathbb{E}^3$ are not EG.
  • fractal geometry (Julia sets, self-affine fractals...) is not EG.
  • not sure about convex bodies. If they're polyhedral I'd say their study fits in EG.
  • packings of spheres are EG.
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9 Answers 9

An important open problem in combinatorial Euclidean geometry is the question of how many different halving lines a set of $2n$ points in the Euclidean plane may have, in the worst case. A halving line is a line through two of the points such that $n-1$ of the points are on each of its sides. The number of halving lines is known to be $O(n^{4/3})$, and there are examples of point sets for which this number is $n2^{\sqrt{\Omega(\log n)}}$, but there remains a large gap between these upper and lower bounds.

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The Unit Distance Problem asks:

For a set of $n$ points in the plane, what is the maximal number $g(n)$ of unit distances realized among the ${n > \choose 2}$ pairs?

A properly scaled square grid gives a lower bound of something like $g(n) \ge n^{1 + \frac{c}{\log \log{n}}}$, and a beautiful application of the crossing number lemma gives that $g(n) = O(n^{4/3})$.

A closely related problem where great progress was made very recently is the Distinct Distance problem, asking for the minimum number $f(n)$ of distinct distances among $n$ points in the plane. (Clearly $f(n)g(n) \ge {n \choose 2}$.)

Guth and Katz recently obtained a sharp exponent for $f(n)$. Terence Tao and János Pach wrote nice summaries of this work.

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I might as well air this question (first posed by Keith Ball) that has sweeping ramifications in convex geometry in high dimensions if the answer is yes:

Let $K$ be a centrally symmetric convex body in $\mathbb{R}^n$, and let $K^\circ$ be the polar or dual convex body. Define a statistic $e(K)$ as the expected value of $(\vec{x} \cdot \vec{y})^2$, where $\vec{x}$ is chosen randomly from $K$ and $\vec{y}$ is chosen randomly from $K^\circ$. Then for each fixed $n$, is $e(K)$ maximized when $K$ is an ellipsoid? The question is even open in two dimensions.

A much weaker conjecture is that the integral of $(\vec{x} \cdot \vec{y})^2$ over $K \times K^\circ$, as opposed to the average value, is maximized when $K$ is an ellipsoid. It is known that $K \times K^\circ$ has the most volume when $K$ is an ellipsoid; this fact is called Santaló's inequality.

It is known that the answer to the first conjecture is no if $K$ is not centrally symmetric, even if the origin is the only points fixed by the symmetries of $K$. (Central symmetry means specifically that $K = -K$.)

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This will certainly be a deep problem, but I'm not sure it fits into the "euclidean geometry" label, as -i guess- a convex body which is not a priori a polytope or an ellipsoid may have a very complicated shape. It sounds more like an 'analysis thing', but someone more knowladgeble than me may judge about this. –  Qfwfq Jan 30 '11 at 23:52
    
If you can prove it for convex polygons in the plane, then I will already be very interested. –  Greg Kuperberg Jan 31 '11 at 10:37
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Moreover, in general convex polytopes are as good as the general case of the problem. –  Greg Kuperberg Jan 31 '11 at 10:37
    
Of course - as you well know, Greg - by the reasoning of your last comment most of the biggest open problems about convex bodies fit here (including the ones I'm surprised you refrained from mentioning in your answer). –  Mark Meckes Jan 31 '11 at 12:58
    
@Mark I decided to go for a question that I particularly care about, instead of offering a general review. –  Greg Kuperberg Jan 31 '11 at 22:29
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The happy ending problem ( http://en.wikipedia.org/wiki/Happy_Ending_problem ) says that any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral. More generally, Erdös and Szekeres proved that for any positive integer $N$, there is a minimal integer $f(N)$ such that any set of $f(N)$ points in the plane in general position has a subset of $N$ points that form the vertices of a convex polygon, and it is known that $f(N)$ is at least $1+2^{N-2}$.

An open question is: does $f(N)=1+2^{N-2}$ hold?

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In recent years there have been a good amount of surveys and publications on "computational" or "combinatorial" geometry, and looking at them may give you a good idea of current questions. Specifically, there is the excellent recent book "Research Problems in Discrete Geometry" by Brass, Moser, and Pach. You may want to start by looking there and at the references it provides. Besides a good deal of information on classical questions, among many other topics, you find:

  • Density problems for packings and coverings.
  • Distance problems.
  • Lattice point problems.
  • Graph drawings and geometric graphs.
  • Geometric inequalities.
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Among the many choices one might get from an Internet search, I suggest Unsolved Problems in Geometry by Hallard Croft, Kenneth Falconer, and Richard Guy (Springer-Verlag, 1991). It may include references to non-Euclidean geometries.

As an aside, I would like to see a geometric proof that the configuration of Pappus is implied by that of Desargues for finite geometric spaces.

Gerhard "Ask Me About System Design" Paseman, 2011.01.30

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Polynomial Hirsch Conjecture?

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In addition to the Croft-Falconer-Guy and Brass-Moser-Pach books others have mentioned, there's Victor Klee and Stan Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, No. 11 in the MAA series, Dolciani Mathematical Expositions, from 1991.

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Let $L_0,L_1,L_2,L_3,L_4$ be five lines in general position on the Euclidean plane---think of the subscripts mod $5$ and draw $L_i$ as the consecutive lines of a (not necessarily regular!) pentagram. Let $C_i$ be the circle inscribed about the triangle formed by $L_i$, $L_{i-2}$, and $L_{i+2}$. Then $C_{i-1}$ and $C_{i+1}$ meet at the intersection of $L_{i-1}$ and $L_{i+1}$, and again at some other point $P_i$ (which we take to be the same point if the two circles are tangent there). Show that these five points $P_i$ are concyclic.
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Actually, it isn't open, but some years ago presented to me as open, with computer-graphical evidence for it...

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