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The proof of the Brunn–Minkowski inequality predates modern measure theory; the development of measure theory and Lebesgue integration allowed connections to be made between geometry and analysis, to the extent that in an integral form of the Brunn–Minkowski inequality known as the Prékopa–Leindler inequality the geometry seems almost ...
Geometry (from Ancient Greek γεωμετρία (geōmetría) 'land measurement'; from γῆ (gê) 'earth, land' and μέτρον (métron) 'a measure') [1] is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. [2]
This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements. [ 21 ] [ 22 ] The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three ...
Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as prime numbers and rational and irrational numbers are introduced. It is proved that there are infinitely many prime numbers. Books XI–XIII concern solid geometry. A typical result is the 1:3 ...
Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. [69] The term was introduced by János Bolyai in 1832. [70] It is sometimes referred to as neutral geometry, [71] as it is neutral with respect to the parallel postulate.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function.