properties of Minkowski’s functional

Let X be a normed space, K convex subset of X and 0 belongs to the interior of K.Then

  1. 1.

    ρK(x)0 for all xX

  2. 2.


  3. 3.

    ρK(λx)=λρK(x), for all λ0 and xX

  4. 4.

    ρK(x+y)ρK(x)+ρK(y) for all x,yK

  5. 5.


  6. 6.

    K0={xX:ρK(x)<1} where K0 denotes the interior of K

  7. 7.

    K¯={xX:ρK(x)1} where K¯ denotes the closureMathworldPlanetmathPlanetmath of K

  8. 8.

    Bd(K)={xX:ρK(x)=1} where the Bd(K) denotes the boundary of K.

Minkowski’s functionalPlanetmathPlanetmathPlanetmath is a useful tool to prove propositionsPlanetmathPlanetmathPlanetmath and solve exercises. Let us see an example
Example Let K be a convex subset of X. Show that Ex(K)Bd(K), where Ex(K) denotes the set of extreme points of K.
If xEx(K) then from this follows that x1K and ρK(x)=1. Now we hypothesize that ρK(x)<1 then there is a real number s such that ρK(x)<s<1 and so ρK(xs)<1. Therefore we have that x=sxs+(1-s)0K, that contradicts to the fact that xEx(K).

Title properties of Minkowski’s functional
Canonical name PropertiesOfMinkowskisFunctional
Date of creation 2013-03-22 15:45:04
Last modified on 2013-03-22 15:45:04
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 10
Author georgiosl (7242)
Entry type Theorem
Classification msc 46B20