Mathematics / advanced
Discrete and convex geometry
Study finite configurations and convex bodies through duality, packing, volume, incidence, and geometric inequalities.
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Module 01 advanced
Discrete and convex geometry
Study finite configurations and convex bodies through duality, packing, volume, incidence, and geometric inequalities.
- 01
Convex polytopes, faces, and f-vectors
Separate a polytope’s metric shape from its face structure and derive consistency checks for face counts.
32 min - 02
Hyperplane arrangements and their cells
Count regions using a deletion-and-restriction recurrence while tracking general-position and degeneracy assumptions.
32 min - 03
Helly, Radon, and Tverberg: finite intersection principles
Connect affine dependence, small intersection certificates, and partitions with intersecting convex hulls.
35 min - 04
Polarity and dual convex bodies
Turn supporting inequalities into dual points while tracking the essential role of the origin and inverse-transpose maps.
32 min - 05
Minkowski sums and mixed volumes
Expand the volume of scaled convex sums and interpret mixed coefficients without confusing them with ordinary intersections.
35 min - 06
The Brunn–Minkowski inequality
Use concavity of the volume root to compare convex sums, identify equality, and understand the first-order isoperimetric consequence.
35 min - 07
Lattice points and Ehrhart polynomials
Count integral dilates of lattice polytopes, interpret reciprocity, and distinguish rational-vertex quasipolynomials.
34 min - 08
Sphere packing and covering bounds
Compute lattice packing and covering densities while separating nonoverlap, coverage, and global optimality claims.
34 min - 09
Incidence geometry and crossing arguments
Convert point–line incidences into graph edges and use the crossing lemma only in its valid density range.
36 min - 10
Geometric discrepancy and range spaces
Balance finite geometric ranges, compare random-coloring bounds with interval structure, and distinguish coloring from sampling.
35 min