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https://math.stackexchange.com/questions/235926/what-is-a-support-function-sup-z-in-k-langle-z-x-rangle
If you take $K$ to be convex, the support function is, in some sense, a tool for a dual representation of the set as the intersection of half-spaces.
https://en.wikipedia.org/wiki/Support_function
The support function is a convex function on . Any non-empty closed convex set A is uniquely determined by h A. Furthermore, the support function, as a function of the set A is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of ...
https://math.stackexchange.com/questions/786765/what-functions-are-support-functions-of-convex-sets
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https://orion.math.iastate.edu/jdhsmith/math/SFGCS.pdf
Support functions of general convex sets Dug-Hwan Choi and Jonathan D. H. Smith Dedicated to the Memory of Gian-Carlo Rota 1. Introduction The concept of the support function of anon-emptycompact convex set was introduced by Minkowski at the end of the 19th century [3, pp. 106, 144, 231].
https://www.encyclopediaofmath.org/index.php/Support_function
A support function is always convex, closed and positively homogeneous (of the first order). The operator is a one-to-one mapping from the family of closed convex sets in onto the family of closed convex homogeneous functions; the inverse operator is the subdifferential (at zero) of the support function.
http://www.stat.cmu.edu/~ryantibs/convexopt-S15/scribes/13-dual-corres-scribed.pdf
We can see that f(x) is the support function of set fzjkzk 1g. We see from the last example that the conjugate of an indicator function is a support function, and the indicator function of a convex set is convex. So the conjugate of a support function is the indictor function. More speci cally, we have: f(y) = I kzk 1(y) 13.3 Lasso Dual
https://en.wikipedia.org/wiki/Convex_function
A twice continuously differentiable function of several variables is convex on a convex set if and only if its Hessian matrix of second partial derivatives is positive semidefinite on the interior of the convex set. Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.
https://www.cs.cmu.edu/~ggordon/10725-F12/scribes/10725_Lecture3.pdf
Lemma 3.8 The function f is convex i the set epi(f) is convex. 3.2.1 Criteria for convexity As with sets, there are multiple ways to characterize a convex function, each of which may by convenient or insightful in di erent contexts. Below we explain the most commonly used three criteriea.
https://see.stanford.edu/materials/lsocoee364b/01-subgradients_notes.pdf
A point x⋆ is a minimizer of a convex function f if and only if f is subdifferentiable at x ... A second and much more difficult task is to describe the complete set of subgradients ∂f(x) as a function of x. We will call this the ‘strong’ calculus of subgradients. It is useful …
https://www.stat.berkeley.edu/~aditya/Site/Research_files/RevisionSuppFunPaperAnnals.pdf
OPTIMAL RATES OF CONVERGENCE FOR CONVEX SET ESTIMATION FROM SUPPORT FUNCTIONS By Adityanand Guntuboyina University of Pennsylvania We present a minimax optimal solution to the problem of esti-mating a compact, convex set from nitely many noisy measurements of its support function. The solution is based on appropriate regular-
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