Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning
نویسندگان
چکیده
منابع مشابه
Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning
The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity. In this paper we present several new results and applications related to pac...
متن کاملShallow Packings in Geometry
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u,v ∈ W is greater than δ, where δ > 0 is an integer...
متن کاملConvexity in SemiAlgebraic Geometry and Polynomial Optimization
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a ba...
متن کاملOn the minimum of a polynomial function on a basic closed semialgebraic set and applications
We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is not zero. As an application, we obtain a lower bound for the separation of two disjoint connected components of basic closed semialgebraic sets, when at least ...
متن کاملTwo Proofs for Shallow Packings
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u,v ∈W is greater than δ, where δ > 0 is an integer ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2019
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-019-00075-0