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The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [ 3 ] [ 4 ] [ 5 ] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph ∪ {\displaystyle \cup } .
According to the above definition, if C is a convex cone, then C ∪ {0} is a convex cone, too. A convex cone is said to be pointed if 0 is in C, and blunt if 0 is not in C. [2] [21] Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β.
In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
Convex polygon, a polygon which encloses a convex set of points; Convex polytope, a polytope with a convex set of points; Convex metric space, a generalization of the convexity notion in abstract metric spaces; Convex function, when the line segment between any two points on the graph of the function lies above or on the graph
[3] [4] Points where concavity changes (between concave and convex) are inflection points. [5] If f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, informally, if the "acceleration" is non-positive). If f ′′ is negative then f is strictly concave, but the converse is not true, as shown by f(x) = −x 4.
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three.
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In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset.
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