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On characterizations of Euclidean spaces Nico Düvelmeyer [email protected] Fakultät für Mathematik TU Chemnitz Germany Educational Workshop on Geometric Inequalities Firenze (Italy), May 2005, 16th-20th N. Düvelmeyer On characterizations of Euclidean spaces 1 Outline 1 Introduction Geometric Background (Banach)-Minkowski Geometry. Well Known Characterization: Parallelogram Equality Well Known Characterization: Symmetry of Orthogonality 2 Area and Arc Length Measure of Angles Defining Measures and Bisectors of Angles Relation Between Area and Arc Length Measure Characterization Generalizing the Symmetry of Orthogonality 3 Further Angular Bisectors Definitions Characterizations N. Düvelmeyer On characterizations of Euclidean spaces 2 Outline 1 Introduction Geometric Background (Banach)-Minkowski Geometry. Well Known Characterization: Parallelogram Equality Well Known Characterization: Symmetry of Orthogonality 2 Area and Arc Length Measure of Angles Defining Measures and Bisectors of Angles Relation Between Area and Arc Length Measure Characterization Generalizing the Symmetry of Orthogonality 3 Further Angular Bisectors Definitions Characterizations N. Düvelmeyer On characterizations of Euclidean spaces 3 Geometric Background of (Banach)-Minkowski Geometry Minkowski Space finite dimensional real linear normed space (finite dimensional Banach space) Md with unit ball B B+x x N. Düvelmeyer 1 y On characterizations of Euclidean spaces 4 Geometric Background of (Banach)-Minkowski Geometry Minkowski Space finite dimensional real linear normed space (finite dimensional Banach space) Md with unit ball B B+x x 1 y Md is Euclidean iff B is an ellipsoid (ellipse). N. Düvelmeyer On characterizations of Euclidean spaces 5 Birkhoffs Orthogonality Relation. |x|B x+Ry x x ay N. Düvelmeyer On characterizations of Euclidean spaces 6 Parallelogram Equality. kx + y k2 + kx − y k2 = 2 kxk2 + 2 ky k2 for all x, y ∈ Md . N. Düvelmeyer On characterizations of Euclidean spaces 7 Symmetry of Orthogonality. Md is Euclidean if dimension d is at least 3, and x a y always implies y a x (x, y ∈ Md ). y+Rx |x|B x+Ry x+Ry y x x =⇒ N. Düvelmeyer On characterizations of Euclidean spaces 8 Symmetry of Orthogonality. Md is Euclidean if dimension d is at least 3, and x a y always implies y a x (x, y ∈ Md ). y+Rx |x|B x+Ry x+Ry y x x =⇒ For d = 2 this property characterizes Radon planes, whose unit circles are Radon curves. N. Düvelmeyer On characterizations of Euclidean spaces 9 Outline 1 Introduction Geometric Background (Banach)-Minkowski Geometry. Well Known Characterization: Parallelogram Equality Well Known Characterization: Symmetry of Orthogonality 2 Area and Arc Length Measure of Angles Defining Measures and Bisectors of Angles Relation Between Area and Arc Length Measure Characterization Generalizing the Symmetry of Orthogonality 3 Further Angular Bisectors Definitions Characterizations N. Düvelmeyer On characterizations of Euclidean spaces 10 Measuring Angles by Arc Length Define measure µl of an angle proportional to the length (measured in Md ) of the corresponding arc of the unit circle (normalized to 2π). N. Düvelmeyer On characterizations of Euclidean spaces 11 Measuring Angles by Arc Length Define measure µl of an angle proportional to the length (measured in Md ) of the corresponding arc of the unit circle (normalized to 2π). This also defines an angular bisector. N. Düvelmeyer On characterizations of Euclidean spaces 12 Measuring Angles by Area Define measure µa of an angle proportional to the area (with arbitrarily chosen unit) of the corresponding sector of the unit circle (normalized to 2π). N. Düvelmeyer On characterizations of Euclidean spaces 13 Measuring Angles by Area Define measure µa of an angle proportional to the area (with arbitrarily chosen unit) of the corresponding sector of the unit circle (normalized to 2π). This also defines an angular bisector. N. Düvelmeyer On characterizations of Euclidean spaces 14 When Are these Two Measures (Length and Area) Identical? Theorem The two measures µa and µl are identical for all angles of some Minkowski plane M2 iff its unit ball is equiframed, i.e., if each point of the unit circle belongs to the boundary of some circumscribed parallelogram of minimal area. N. Düvelmeyer On characterizations of Euclidean spaces 15 When Are these Two Measures (Length and Area) Identical? Theorem The two measures µa and µl are identical for all angles of some Minkowski plane M2 iff its unit ball is equiframed, i.e., if each point of the unit circle belongs to the boundary of some circumscribed parallelogram of minimal area. N. Düvelmeyer On characterizations of Euclidean spaces 16 When Are these Two Measures (Length and Area) Identical? Theorem The two measures µa and µl are identical for all angles of some Minkowski plane M2 iff its unit ball is equiframed, i.e., if each point of the unit circle belongs to the boundary of some circumscribed parallelogram of minimal area. N. Düvelmeyer On characterizations of Euclidean spaces 17 When Are these Two Measures (Length and Area) Identical? Theorem The two measures µa and µl are identical for all angles of some Minkowski plane M2 iff its unit ball is equiframed, i.e., if each point of the unit circle belongs to the boundary of some circumscribed parallelogram of minimal area. Especially, this holds for all planes with symmetric orthogonality (Radon planes). N. Düvelmeyer On characterizations of Euclidean spaces 18 New Result. Theorem A Minkowski space Md (d ≥ 3) is Euclidean iff for each two-dimensional subspace the unit disc is equiframed. N. Düvelmeyer On characterizations of Euclidean spaces 19 Sketch of the Proof reduce to the case of symmetric orthogonality uses reductio ad absurdum local difference with adjacent straight segments in the unit circle extends to planar part in three-dimensional unit ball this subconfiguration has no end (unbounded cylinder) N. Düvelmeyer On characterizations of Euclidean spaces 20 Outline 1 Introduction Geometric Background (Banach)-Minkowski Geometry. Well Known Characterization: Parallelogram Equality Well Known Characterization: Symmetry of Orthogonality 2 Area and Arc Length Measure of Angles Defining Measures and Bisectors of Angles Relation Between Area and Arc Length Measure Characterization Generalizing the Symmetry of Orthogonality 3 Further Angular Bisectors Definitions Characterizations N. Düvelmeyer On characterizations of Euclidean spaces 21 Other Angular Bisectors. Further properties of angular bisectors in the Euclidean plane can be used to define Angular Bisectors: set of points equidistant to the sides (Glogovskij) N. Düvelmeyer On characterizations of Euclidean spaces 22 Other Angular Bisectors. Further properties of angular bisectors in the Euclidean plane can be used to define Angular Bisectors: b a b d = c d a c set of points equidistant to the sides (Glogovskij) N. Düvelmeyer The ray dividing each secant in the ratio of the lengths of corresponding segments on the sides. (Busemann) On characterizations of Euclidean spaces 23 Characterizations Using Equivalent Systems of Angular Bisectors in the Plane ≡ Glogovskij measure µ2 Busemann measure µ1 N. Düvelmeyer On characterizations of Euclidean spaces 24 Characterizations Using Equivalent Systems of Angular Bisectors in the Plane ≡ Glogovskij Busemann iff M2 is Radon measure µ2 measure µ1 N. Düvelmeyer On characterizations of Euclidean spaces 25 Characterizations Using Equivalent Systems of Angular Bisectors in the Plane ≡ Glogovskij measure µ2 Busemann iff M2 is Radon iff M2 is Euclidean and µ2 = µa = µl measure µ1 N. Düvelmeyer On characterizations of Euclidean spaces 26 Characterizations Using Equivalent Systems of Angular Bisectors in the Plane ≡ Glogovskij measure µ2 Busemann iff M2 is Radon iff M2 is Euclidean and µ2 = µa = µl measure µ1 iff M2 is Euclidean and µ1 = µa = µl N. Düvelmeyer On characterizations of Euclidean spaces 27 Characterizations Using Equivalent Systems of Angular Bisectors in the Plane ≡ Glogovskij measure µ2 Busemann iff M2 is Radon iff M2 is Euclidean and µ2 = µa = µl measure µ1 iff M2 is Euclidean iff µ1 = µ2 and µ1 = µa = µl N. Düvelmeyer On characterizations of Euclidean spaces 28 Characterizations Using Equivalent Systems of Angular Bisectors in the Plane ≡ Glogovskij measure µ2 Busemann iff M2 is Radon iff M2 is Euclidean and µ2 = µa = µl measure µ1 iff M2 is Euclidean iff µ1 = µ2 and µ1 = µa = µl for µl = µa : iff M2 has an equiframed unit circle N. Düvelmeyer On characterizations of Euclidean spaces 29 Characterizations Using Equivalent Systems of Angular Bisectors in Md , d ≥ 3. ≡ Glogovskij measure µ2 iff Md is Euclidean Busemann and µ2 = µa = µl measure µ1 iff Md is Euclidean and µ1 = µa = µl N. Düvelmeyer On characterizations of Euclidean spaces 30 Characterizations Using Equivalent Systems of Angular Bisectors in Md , d ≥ 3. ≡ Glogovskij measure µ2 Busemann iff Md is Euclidean iff Md is Euclidean and µ2 = µa = µl measure µ1 iff Md is Euclidean and µ1 = µa = µl N. Düvelmeyer On characterizations of Euclidean spaces 31 Characterizations Using Equivalent Systems of Angular Bisectors in Md , d ≥ 3. ≡ Glogovskij measure µ2 Busemann iff Md is Euclidean iff Md is Euclidean and µ2 = µa = µl measure µ1 N. Düvelmeyer iff Md is Euclidean iff µ1 = µ2 and µ1 = µa = µl for µl = µa : iff Md is Euclidean On characterizations of Euclidean spaces 32 Summary There are a lot of properties which characterize Euclidean spaces within the family of Minkowski spaces. They can be regarded as proofs that our world is “the best of all possible worlds...” (3-dimensional space) We have seen four new such characterizations. N. Düvelmeyer On characterizations of Euclidean spaces 33 For Further Reading Nico Düvelmeyer. A new characterization of Radon curves via angular bisectors. Journal of Geometry, 80(1–2):75 – 81, 2004. Nico Düvelmeyer. Angle measures and bisectors in Minkowski planes. Canadian Mathematical Bulletin, to appear. Nico Düvelmeyer. On convex bodies all whose two-dimensional sections are equiframed. to be submitted N. Düvelmeyer On characterizations of Euclidean spaces 34