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In geometry, a disk (also spelled disc) [1] is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not. [2] For a radius, , an open disk is usually denoted as and a closed disk is ¯.
The spherical isoperimetric inequality states that (), and that the equality holds if and only if the curve is a circle. There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.
This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed.
A closed rectangle V is not a neighbourhood of any of its corners or its boundary since there is no open set in V containing any corner or edge point. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points.
The closed disk is a simple example of a surface with boundary. The boundary of the disc is a circle. The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface.
Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is: A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.
In mathematics, the Poincaré inequality [1] is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition.
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.