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The light gray area is the absolutely convex hull of the cross. A subset S {\displaystyle S} of a real or complex vector space X {\displaystyle X} is called a disk and is said to be disked , absolutely convex , and convex balanced if any of the following equivalent conditions is satisfied:
The Banach space (X, ǁ ⋅ ǁ) is a strictly convex space (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2.
The function () = has ″ = >, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function () = has ″ =, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points.
This occurs between water and glass. Water-based fluids like sap, honey, and milk also have a concave meniscus in glass or other wettable containers. Conversely, a convex meniscus occurs when the adhesion energy is less than half the cohesion energy. Convex menisci occur, for example, between mercury and glass in barometers [1] and thermometers.
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
The convex layers of a point set and their intersection with a halfplane. In computational geometry, the convex layers of a set of points in the Euclidean plane are a sequence of nested convex polygons having the points as their vertices. The outermost one is the convex hull of the points and the rest are formed in the same way recursively.
The convex hull of the extreme points of forms a convex subset of so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of . For this reason, the following corollary to the above theorem is also often called the Krein–Milman theorem.
For instance, the traveling salesman problem, NP-hard for arbitrary sets of points in the plane, is trivial for points in convex position: the optimal tour is the convex hull. [3] Similarly, the minimum-weight triangulation of planar point sets is NP-hard for arbitrary point sets, [ 4 ] but solvable in polynomial time by dynamic programming for ...