If is a convex set, for any in, and any nonnegative numbers such that, then the vector

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

If binding around a not fully convex, or square-edged object, arrange the knot so the overhand knot portion is stretched across a convex portion, or a corner, with the riding turn squarely on top of it.

If the matrix is positive semidefinite matrix, then is a convex function: In this case the quadratic program has a global minimizer if there exists some feasible vector ( satisfying the constraints ) and if is bounded below on the feasible region.

If the points of are all on a line, the convex hull is the line segment joining the outermost two points.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

If the gradient is less ( as it almost always is ) the rays are not bent enough and get lost in space, which is the normal situation of a spherical, convex " horizon ".

If a convex ( i. e., concave upward ) function has a " bottom ", any point at the bottom is a minimal extremum.

If g is a real-valued function that is μ-integrable, and if is a convex function on the real line, then:

If a region is not convex, a " dent " in its boundary can be " flipped " to increase the area of the region while keeping the perimeter unchanged.

If a kernel is non-negative, such as for a Gaussian kernel, then the value of the filtered signal will be a convex combination of the input values ( the coefficients ( the kernel ) integrate to 1, and are non-negative ), and will thus fall between the minimum and maximum of the input signal – it will not undershoot or overshoot.

If F < sup > 2 </ sup > is strongly convex, then F is a Minkowski norm on each tangent space.

If it is a " right turn ", this means that the second-to-last point is not part of the convex hull and should be removed from consideration.

( If at any stage the three points are collinear, one may opt either to discard or to report it, since in some applications it is required to find all points on the boundary of the convex hull.

* Jensen's inequality holds: If ƒ is a convex function, then

If S is a polytope, then the k-extreme points are exactly the interior points of the k-dimensional faces of S. More generally, for any convex set S, the k-extreme points are partitioned into k-dimensional open faces.

If S is closed, bounded, and n-dimensional, and if p is a point in S, then p is k-extreme for some k < n. The theorem asserts that p is a convex combination of extreme points.

If instead one restricts to all maps with given endpoint values: such that and, then for the inclusion of functions with non-vanishing derivative in all continuous functions is a homotopy equivalence – both the spaces are convex, and in fact the monotone functions are a convex subset.

If every internal angle of a simple, closed polygon is less than 180 °, the polygon is called convex.

If the object being bounded is known to be convex, this is not a restriction.

If non-convex bounding volumes are required, an approach is to represent them as a union of a number of convex bounding volumes.

If the object is the union of a finite set of points, its convex hull is a polytope.

If a ship's hull classification symbol begins with " W ", it is a ship of the United States Coast Guard.

If a ship's hull classification symbol begins with " T -", it is part of the Military Sealift Command, has a primarily civilian crew, and is a United States Naval Ship ( USNS ) in non-commissioned service — as opposed to a commissioned United States Ship ( USS ).

If a meteoroid penetrated the satellite's outer hull, it would be detected by the temperature data sent back to Earth.

If the situation is reversed, with the center of pressure forward of the center of lateral resistance of the hull, a " lee " helm will result, which is generally considered undesirable, if not dangerous.

If heated too quickly, the steam in the outer layers of the kernel can reach high pressures and rupture the hull before the starch in the center of the kernel can fully gelatinize, leading to partially popped kernels with hard centers.

If damaged " between wind and water " ( i. e., in the exposed section of the hull ) she was consequently in danger of sinking when on the other tack.

If the length of the watercraft deck prevents the mounting of three masts, it is a boat, while a single-masted hull can be termed a craft.

If f is a H-essential H-morphism with a domain X and an H-injective codomain G, G is called an H-injective hull of X.

If steel is used, a zinc layer is often applied to coat the entire hull.

If the spinnaker chute penetrates the hull and is required to be watertight, it takes the form of a hard tube sealed to the hull at both ends.

If C is locally small, satisfies Grothendieck's axiom AB5 ) and has enough injectives, then every object in C has an injective hull ( these three conditions are satisfied by the category of modules over a ring ).

If built within the hull, rather than forming the outer hull, the belt would be installed at an inclined angle to improve the warships protection from shells striking the hull.

One has to construct the convex hull of the set S < sub > 1 </ sub > and project it back onto the space of S. If points are not in general position, additional effort is required to triangulate the non-tetrahedral facets.

If built within the hull, rather than forming the outer hull it could be fitted at an inclined angle to improve the protection.

If it is not, there is guaranteed to exist a linear inequality that separates the optimum from the convex hull of the true feasible set.

If the boat is righted from a position where the mast is to some degree upwind of the hull, the boat ’ s inertia, coupled with the wind pressure, can cause it to immediately capsize in the opposite direction.

* If numbers have mean X, then.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* Theorem If X is a normed space, then X ′ is a Banach space.

If X ′ is separable, then X is separable.

If F is also surjective, then the Banach space X is called reflexive.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

If there is a bounded linear operator from X onto Y, then Y is reflexive.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.