Convex bodies: the brunn–minkowski theory

Advances in mathematics

THIS is a thoroughly good book which deserves to run through many editions. It is not (as its title might suggest) a book on mathematical logic or philosophy: it deals not with the nature of mathematics but with its content. Its purpose is to show, not by general disquisitions but by concrete examples, drawn from almost every branch of pure mathematics, how mathematicians think and what they do. What is Mathematics? An Elementary Approach to Ideas and Methods. By Richard Courant and Herbert Robbins. Pp. xix + 521. (London, New York and Toronto: Oxford University Press, 1941.) 25s. net.

What is Mathematics

Let X be a space and F a family of 0, 1-valued functions on X. Vapnik and Chervonenkis showed that if F is "simple" (finite VC dimension), then for every probability measure μ on X and e > 0 there is a finite set S such that for all f e F, Σxesf(x)/|S| = |f f(x)dμ(x)] ± e.Think of S as a "universal e-approximator" for integration in F. S can actually be obtained w.h.p. just by sampling a few points from μ. This is a mainstay of computational learning theory. It was later extended by other authors to families of bounded (e.g., [0, 1]-valued) real functions.In this work we establish similar "universal e-approximators" for families of unbounded nonnegative real functions --- in particular, for the families over which one optimizes when performing data classification. (In this case the e-approximation should be multiplicative.)Specifically, let F be the family of "k-median functions" (or k-means, etc.) on Rd with an arbitrary norm ϱ. That is, any set u1,..., uk e Rd determines an f by f(x) = (mini ϱ(x - ui))α. (Here α ≥ 0.) Then for every measure μ on Rd there exists a set S of cardinality poly(k, d, 1/e) and a measure v supported on S such that for every f e F, Σxesf(x)v(x) e (1 ± e) · (f f (x)dμ(x)).

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Universal ε-approximators for integrals

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Advances in Applied Mathematics

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.

Ball-Polyhedra

We study two notions One is that of spindle convexity A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra We find analogues of several results on convex polyhedral sets for ball-polyhedra

Barany, Katchalski and Pach (Proc Am Math Soc 86(1):109---114, 1982) (see also Barany et al., Am Math Mon 91(6):362---365, 1984) proved the following quantitative form of Helly's theorem. If the intersection of a family of convex sets in $$\mathbb {R}^d$$Rd is of volume one, then the intersection of some subfamily of at most 2d members is of volume at most some constant v(d). In Barany et al. (Am Math Mon 91(6):362---365, 1984), the bound $$v(d)\le d^{2d^2}$$v(d)≤d2d2 was proved and $$v(d)\le d^{cd}$$v(d)≤dcd was conjectured. We confirm it.

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Proof of a Conjecture of Bárány, Katchalski and Pach

Rigidity of ball-polyhedra in Euclidean 3-space

We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean n-space with translates of a convex body, or more generally, any measurable set. We obtain a bound for the density of covering the n-sphere by rotated copies of a spherically convex set (or, any measurable set). Using the same method, we sharpen an estimate by Artstein-Avidan and Slomka on covering a bounded set by translates of another. The main novelty of our method is that it is not probabilistic. The key idea, which makes our proofs rather simple and uniform through different settings, is an algorithmic result of Lovasz and Stein. © 2016 American Mathematical Society.

Márton Naszódi

Papers

Ball-Polyhedra

Ball-Polyhedra

Proof of a Conjecture of Bárány, Katchalski and Pach

Rigidity of ball-polyhedra in Euclidean 3-space

On some covering problems in geometry