Transference Theorems in the Geometry of Numbers Daniel

Transference Theorems in the Geometry of Numbers Daniel Dadush New York University EPIT 2013

Convex Bodies Non convex set.

Integer Programming Problem (IP) convex set •

Integer Programming Problem (IP) Linear Programming (LP) convex set •

Integer Programming Problem (IP) •

Integer Programming Problem (IP) •

Central Geometric Questions 1) When can we guarantee that a convex set contains a lattice point? (guarantee IP feasibility) 2) What do lattice free convex sets look like? (sets not containing integer points)

Examples “Hidden cube” If a convex set very ``fat’’, then it will always contain a lattice point.

Examples If a convex set very ``fat’’, then it will always contain a lattice point.

Examples Infinite band Volume does NOT guarantee lattice points (in contrast with Minkowski’s theorem).

Examples However, lattice point free sets must be ``flat’’ in some direction.

Lattice Width • …

Lattice Width •

Lattice Width •

Lattice Width subproblems •

Lattice Width subproblems •

Lattice Width •

Lattice Width •

Lattice Width •

Lattice Width •

Lattice Width •

Kinchine’s Flatness Theorem •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Convex & Centrally Symmetric •

Kinchine’s Flatness Theorem •

Kinchine’s Flatness Theorem •

Kinchine’s Flatness Theorem •

Kinchine’s Flatness Theorem •

Kinchine’s Flatness Theorem •

Lower Bound: Simplex • No interior lattice points.

Lower Bound: Simplex •

Flatness Theorem •

Flatness Theorem •

Covering Radius •

Covering Radius •

Covering Radius • Condition from Flatness Theorem

Covering Radius • Condition from Flatness Theorem

Covering Radius •

Covering Radius •

Covering Radius •

Covering Radius •

Covering Radius •

Flatness Theorem •

Flatness Theorem •

Flatness Theorem •

Flatness Theorem •

Flatness Theorem •

Lower Bounds: Random Lattices •

Norms and Convex Bodies

Norms and Convex Bodies

Norms and Convex Bodies convex combination

Norms and Convex Bodies

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem

Successive Minima

Successive Minima

Successive Minima

Minkowski’s First Theorem

Minkowski’s First Theorem

Minkowski’s First Theorem

Minkowski’s Second Theorem

Covering Radius vs Successive Minima “Naïve” Babai rounding

Separator Theorem

Separator Theorem

Polar Bodies and Dual Norms (-1, 1) (0, 1) (1, 1) (-1, 0) (-1, -1) (1, 0) (0, -1)

Polar Bodies and Dual Norms

Polar Bodies and Dual Norms

Polar Bodies and Dual Norms

Polar Bodies and Dual Norms

Banaszczyk’s Transference Theorem

Volume Product Bounds

Minkowski Transference

Projected Norms and Lattices

Flatness Theorem •

Flatness Theorem •

Generalized Babai Rounding •

Generalized Babai Rounding •

Generalized Babai Rounding •

Generalized Babai Rounding •

Generalized Babai Rounding •

Generalized Babai Rounding •

Generalized Babai Rounding •

Generalized Babai Rounding •

Generalized HKZ Basis •

Flatness Theorem •

Flatness Theorem •

Flatness Theorem •

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Subspace Flatness Theorem

Inhomogeneous Minkowski •

Inhomogeneous Minkowski •

Gaussian Heuristic •

Subspace Flatness •

Subspace Flatness •

Generalized HKZ Basis •

Inhomogeneous Minkowski •

Inhomogeneous Minkowski •

Inhomogeneous Minkowski •

Gaussian Heuristic •

Gaussian Heuristic

Gaussian Heuristic

Gaussian Heuristic

Gaussian Heuristic

Subspace Flatness Theorem •

Subspace Flatness

Subspace Flatness

Subspace Flatness

Subspace Flatness

Subspace Flatness Theorem •