1、詹森不等式以丹麦数学家约翰詹森（Johan Jensen）命名。它给出积分的凸函数值和凸函数的积分值间的关系。 Jensens inequality generalizes the statement that a secant line of a convex function lies above the graph.詹森不等式的一般形式詹森不等式可以用测度论或概率论的语言给出。这两种方式都表明同一个很一般的结果。 测度论的版本假设 是集合 的正测度，使得 () = 1。若 g 是勒贝格可积的实值函数，而 是在 g 的值域上定义的凸函数，则 。 概率论的版本以概率论的名词， 是个概率测度。
2、函数 g 换作实值随机变量 X（就纯数学而言，两者没有分别）。在 空间上，任何函数相对于概率测度 的积分就成了期望值。这不等式就说，若是任一凸函数，则 。 詹森不等式的特例机率密度函数的形式假设 是实数轴上的可测子集，而 f(x)是非负函数，使得 。 以概率论的语言，f 是个机率密度函数。 詹森不等式变成以下关于凸积分的命题： 若 g 是任一实值可测函数， 在 g 的值域中是凸函数，则 。 若 g(x) = x，则这形式的不等式简化成一个常用特例： 。 有限形式若 是有限集合 ，而 是 上的正规计数测度，则不等式的一般形式可以简单地用和式表示： ， 其中 。 若 是凹函数，只需把不等式符号调转
3、。 假设 是正实数，g(x) = x， i = 1 / n 及 。上述和式便成了 ， 两边取自然指数就得出熟悉的平均数不等式： 。 这不等式也有无限项的离散形式。 统计物理学统计物理学中，若凸函数是指数函数，詹森不等式特别重要： ， 其中方括号表示期望值，是以随机变量 X 的某个概率分布 算出。这个情形的证明很简单（参见 Chandler, Sec. 5.5）：在以下等式的第三个指数函数 套用不等式 ， 詹森不等式(Jensens inequity)证明方法备忘录詹森不等式(Jensens inequity)证明方法备忘录若 f 为凹函数，即 f=a1*f(x1)+a2*f(x2)+an*xn
4、恒成立。证明：不是一般性，令 xi=a1*f(x1)+(1-a1)*f(x2)成立欲证上式成立，即证明a1+(1-a1) *fa1*x1+(1-a1)*x2 =a1*f(x1)+(1-a1)*f(x2)成立，移项并合并同类项后上式可变为：a1*fa1*x1+(1-a1)*x2- f(x1)=(1-a1)* f(x2)- fa1*x1+(1-a1)*x2 （1）根据罗尔中值定理，有：f(x)-f(x)/(x-x)=f(y),x=(1-a1)*f(y2)*a1*(x2-x1),其中 y2=y1。由 f=f(y2)，因此（1）式成立。 （注：麦克劳林展开只有在（x2-x1)趋于零时成立，因此，此处不
5、能使用麦克劳林展开公式） 。（2）证明 n=2k(k 为正整数)时命题成立，首先证明 n=4 时原命题成立，则由（1）可得：（a1+a2）*fa1/(a1+a2)*x1+a2/(a1+a2)*x2=(a1+a2)*a1/(a1+a2)f(x1)+a2/(a1+a2)*f(x2) (2)（a3+a4）*fa3/(a3+a4)*x3+a4/(a3+a4)*x4=(a3+a4)*a3/(a3+a4)f(x3)+a4/(a3+a4)*f(x4) (3)将上述（2）式与（3）式同时除以（a1+a2+a3+a4） ，再次利用（1）式可得：fa1/(a1+a2+a3+a4)*x1+a2/( a1+a2+a3
6、+a4)*x2+a3/( a1+a2+a3+a4)*x3+ a4/( a1+a2+a3+a4)*x4=(a1+a2)/(a1+a2+a3+a4)* fa1/(a1+a2)*x1+a2/(a1+a2)*x2+（a3+a4）/(a1+a2+a3+a4)*fa3/(a3+a4)*x3+a4/(a3+a4)*x4(4)由于 a1+a2+a3+a4=1，因此（4）式左边部分即为 f(a1*x1+a2*x2+a3*x3+a4*x4)，右边部分即为 a1*f(x1)+a2*f(x2)+a3*f(x3)+a4*f(x4)，n=4 时，命题得证。同理可从最底层开始运用公式（1）证明 n=2k(k2)的情形依然成
7、立。（3）当 n=Ef(x)，将 f 函数符号改为 U 符号，即得微观经济学中冯诺依曼期望效用的一个不等式。DefinitionAn affine space is a set together with a vector space and a group action of (with addition of vectors as group operation) on , such that the only vector acting with a fixpoint is (i.e., the action is free) and there is a single orbit (t
8、he action is transitive). In other words, an affine space is a principal homogeneous space over the additive group of a vector space.Explicitly, an affine space is a point set together with a mapwith the following properties:1. Left identity2. Associativity3. Uniquenessis a bijection.The vector spac
9、e is said to underlie the affine space and is also called the difference space.By choosing an origin, , one can thus identify with , hence turn into a vector space. Conversely, any vector space, , is an affine space over itself. The uniquenessproperty ensures that subtraction of any two elements of
10、is well defined, producing a vector of .If , , and are points in and is a scalar, thenis independent of . Instead of arbitrary linear combinations, only such affine combinations of points have meaning.By noting that one can define subtraction of points of an affine space as follows:is the unique vec
11、tor in such that ,one can equivalently define an affine space as a point set , together with a vector space , and a subtraction map with the following properties 2:1. there is a unique point such that and2. .These two properties are called Weyls axioms.editExamples When children find the answers to
12、sums such as 4+3 or 42 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space. Any coset of a subspace of a vector space is an affine space over . If is a matrix and lies in its column space, the set of solutions of the equation is an affine s
13、pace over the subspace of solutions of . The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation. Generalizing all of the above, if is a linear mapping and y lies in its image, the set of solutions to the
14、 equation is a coset of the kernel of , and is therefore an affine space over .editAffine subspacesAn affine subspace (sometimes called a linear manifold, linear variety, or a flat) of a vector space is a subset closed under affine combinations of vectors in the space. For example, the setis an affi
15、ne space, where is a family of vectors in this space is the affine span of these points. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace of This affine subspace can be equivalently described as the coset of the -actionwhere is any
16、element of , or equivalently as any level set of the quotient map A choice of gives a base point of and an identification of with but there is no natural choice, nor a natural identification of with A linear transformation is a function that preserves all linear combinations; an affine transformatio
17、n is a function that preserves all affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.For example, in , the origin, lines and planes through the origin and the whole space are linear subspaces, whi
18、le points, lines and planes in general as well as the whole space are the affine subspaces.editAffine combinations and affine dependenceMain article: Affine combinationAn affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are lin
19、early independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their “linear span“ and is always a linear subspace; the set of all affine combinations is thei
20、r “affine span“ and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three.Vectorsv1, v2, ., vnare linearly dependent if there exist scalars a1, a2,
21、,an, not all zero, for whicha1v1 + a2v2 + + anvn = 0(1)Similarly they are affinely dependent if in addition the sum of coefficients is zero:editAxiomsAffine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic geometry by
22、 writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.Coxeter (1969, p.192) axiomatizes affine geometry (over the reals) as ordered geometry together with an affine form of Desarguess theorem and an axiom stating that in a plan
23、e there is at most one line through a given point not meeting a given line.Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint): Any two distinct points lie on a unique line. Given a point and line there is a uni
24、que line which contains the point and is parallel to the line There exist three non-collinear points.As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher dimensional affine spa
25、ces.editRelation to projective spacesAn affine space is a subspace of projective space, which is in turn a quotient of a vector space.Affine spaces are subspaces of projective spaces: an affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversel
26、y any affine plane can be used to construct a projective plane as a closure by adding a “line at infinity“ whose points correspond to equivalence classes of parallel lines.Further, transformations of projective space that preserve affine space (equivalently, that preserve the points at infinity as a
27、 set) yield transformations of affine space, and conversely any affine linear transformation extends uniquely to a projective linear transformations, so affine transformations are a subset of projective transforms. Most familiar is that Mbius transformations (transformations of the projective line,
28、or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.However, one cannot take the projectivization of an affine space, so projective spaces are not naturally quotients of affine spaces: one can only take the projectivization of a vector space, since the projective space is lines through a given point, and there is no distinguished point in an affine space. If one chooses a base point (as zero), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.
