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Convexity and Related Classic Inequalities

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Abstract

Concepts of convexity and related techniques are widely used in various branches of mathematics and in applications such as functional analysis, calculus of variations, and mathematical physics, geometry, number theory, theory of games, optimization and other computational methods and convex programming, integral geometry (tomography), and many others. In this chapter we will focus mainly on the basic elements of the theory of convex functions, which can be taught to readers with limited experience. The remaining problems are directed toward specific applications such as convex and linear programming, hierarchy of power means, geometric inequalities, the Hölder and Minkowski inequalities, Young’s inequality and the Legendre transform, and functions on metric and normed vector spaces. The necessary introductory material is provided in these sections. A specially added section discusses the widely known Cauchy-Schwarz-Bunyakovskii (CSB) inequality and Lagrange-type identities in connection with the hierarchy of the power means.

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Notes

  1. 1.

    This definition allows disconnected polyhedrons and unions of polyhedrons intersecting along edges of codimensions greater than one. Examples are polyhedrons with a common vertex and polyhedrons obtained by partitioning faces (with the corresponding partition and addition of edges of all dimensions) into smaller ones, which would count as different.

  2. 2.

    Proving these claims, obvious for planar polygons, requires more sophisticated methods for dimensions greater than two. Working on this problem will help the reader in learning those methods because most real-life problems are of high dimensions.

  3. 3.

    For advanced readers, the proper name for these shapes is n-dimensional submanifolds with piecewise-planar boundaries.

  4. 4.

    The first example is a special case of the second one.

  5. 5.

    In an infinite-dimensional case, the hyperplanes of support are required to be closed.

  6. 6.

    Actually, there are discontinuous linear functionals on any topological vector space (a vector space over either the real or the complex field of scalars supplied with a topology so that with respect to it the summation of vectors and multiplication of them by scalars become continuous operations) possessing a base of neighborhoods of zero of a cardinality not exceeding the algebraic dimension of the space. Indeed, following sources such as Kirillov and Gvishiani (1988), let U \( \mapsto \) e U be an injective map on this base to any fixed basis of our space and k U be positive numbers such that k U e U ∈ U, ∀U (k U cannot be chosen without the famous axiom of choice, but of course we accept it). A linear functional f taking values k U −1 on the corresponding e U and arbitrary values on the rest of the basis is discontinuous since the set {f < 1}, which is the preimage of (an open) neighborhood of zero in ℝ, includes no U and so cannot be a neighborhood of zero in our space. [We leave it to readers familiar with the elements of general topology to work out the details, including the fact that any neighborhood of zero in a topological vector space contains vectors of all directions (in fact, contains some half-interval [0, α v v) of the rectilinear ray spanned on any vector v (α v  > 0)), which follows from the continuity of the multiplication by scalars. We must warn readers, however, that the definition of a topological vector space uses a linguistic trick: a vector space of positive dimension with discrete topology (which means that every subset is open) is a topological Abelian group (with respect to the summation of vectors) but is not a topological vector space.]

  7. 7.

    This theorem, due to I.M. Gelfand, is analogous to the Banach-Steinhaus theorem for linear operators.

  8. 8.

    Advanced readers probably know that in infinite-dimensional analysis, compactness does not mean the same as boundedness + closure. A topological space is compact if and only if every infinite subset has a limit point. Readers may prove this using the common definition of compactness that they are familiar with or find it in a textbook on general topology, for example, Hausdorff (1914) or Bourbaki (1960). Use this equivalent definition to prove the statement about the compactness of the convex hull of a compact set.

  9. 9.

    Readers may provide examples.

  10. 10.

    Hyperplanes F 1,…,F m are in a general position when codim ⋂F i  = m, or, in other words (why?), the vectors orthogonal to F i are linearly independent. Among the faces containing a vertex, there are always n faces in a general position, because otherwise the intersection would have a positive dimension and not be a vertex.

  11. 11.

    Actually, those definitions are dual with each other, as discussed in section H9.5, so their equivalence means self-duality (autoduality) of the simplicial shape.

  12. 12.

    A similar result (without exact estimates) may be obtained by using the continuity of a scalar product.

  13. 13.

    The famous Hahn-Banach theorem states, in one of its geometrical versions, that a closed convex set has hyperplanes of support at all its face points. It can be proved with no regard for the compactness, which allows multiple applications in infinite-dimensional analysis, where boundedness + closure ≠ compactness. (In infinite-dimensional problems, mostly closed hyperplanes of support are considered.) For this reason, the Hahn-Banach theorem is usually learned in courses on functional analysis. Readers seeking more information may consult Dunford and Schwartz (1957), Hille and Phillips (1957), Yosida (1965), (Riesz and Nagy 1972), Edwards (1965), Reed and Simon (1972), Rudin (1973), Kolmogorov and Fomin (1976), Rockafellar (1970), and Kutateladze and Rubinov (1976).

  14. 14.

    Advanced readers probably know that this statement is generalized for any analytic function (a one, representable by the sum of a convergent power series in a neighborhood of any point of its domain): an analytic function on an open connected set will be equal to zero on the entire domain if it equals zero on its open subset. Interested readers will find more details in Dieudonné (1960).

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  1. Rishon LeZion, Israel

    Alexander A. Roytvarf

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Roytvarf, A.A. (2013). Convexity and Related Classic Inequalities. In: Thinking in Problems. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8406-8_9

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