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Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
By the Minkowski inequality, the function h K+ p L is again positive homogeneous and convex and hence the support function of a compact convex set. This definition is fundamental in the L p Brunn-Minkowski theory.
Every norm, seminorm, and real linear functional is a sublinear function.The identity function on := is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation . [5] More generally, for any real , the map ,: {is a sublinear function on := and moreover, every sublinear function : is of this ...
Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map : is called a sublinear function if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is not necessarily nonnegative.
Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree . As with the phase function ϕ {\displaystyle \phi } , in some cases the function a {\displaystyle a} is taken to be in more general, or just different, classes.
the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and; the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).
Several proofs are known, [3] one is using the fact that the Legendre transform of a positive homogeneous, convex, real valued function is the (convex) indicator function of a compact convex set. Many authors restrict the support function to the Euclidean unit sphere and consider it as a function on S n-1.
More generally, if is convex and the origin belongs to the algebraic interior of , then is a nonnegative sublinear functional on , which implies in particular that it is subadditive and positive homogeneous.