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The isoperimetric inequality states that , and that the equality holds if and only if the curve is a circle. The area of a disk of radius R is πR 2 and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π 2 R 2 in this case.
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .
A simple procedure to determine which half-plane is in the solution set is to calculate the value of ax + by at a point (x 0, y 0) which is not on the line and observe whether or not the inequality is satisfied. For example, [3] to draw the solution set of x + 3y < 9, one first draws the line with equation x + 3y = 9 as a dotted line, to ...
A simple integration over r (or sum in the case of graphs) shows that a d-dimensional isoperimetric inequality implies a d-dimensional volume growth, namely (,) where B(x,r) denotes the ball of radius r around the point x in the Riemannian distance or in the graph distance. In general, the opposite is not true, i.e. even uniformly ...
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
Proof [2]. Since + =, =. A graph = on the -plane is thus also a graph =. From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines =, =, =, =, and the fact that is always increasing for increasing and vice versa, we can see that upper bounds the area of the rectangle below the curve (with equality ...
A cycle graph in which the distances disobey Ptolemy's inequality. Ptolemy's inequality holds more generally in any inner product space, [1] [9] and whenever it is true for a real normed vector space, that space must be an inner product space. [9] [10] For other types of metric space, the inequality may or may not be
In the context of linear programming and related problems in mathematical optimization, convex polytopes are often described by a system of linear inequalities that their points must obey. When a polytope is integral, linear programming can be used to solve integer programming problems for the given system of inequalities, a problem that can ...