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What 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.
• tessellations 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.

What 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 defined 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.
• tessellations 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.

What 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.

What 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.
• tessellations 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.

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.

What 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.