Convex geometry and Brunn-Minkowski theory 1000-1M25GWBM
1. Convex sets in ����ⁿ.
a) Description by convex combinations and as intersection of a family of half-spaces
b) Polar body
c) Carathéodory, Helly and Radon theorems
d) Topological properties: relative interior and boundary
e) Separation theorems
f) Extreme and exposed points
2 Convex functions.
a) Epigraph
b) Convex conjugates
c) Operations on convex functions (e.g. inf-convolution)
d) Regularity and existence of one-sided directional derivatives
e) Support function and its conjugate
3) The metric space of convex sets
a) Metrics on convex sets
b) Blaschke's selection theorem
c) Polyhedral approximation
4. Brunn-Minkowski theory
a) Volume and surface measure
b) Mixed-volumes
c) Quermassintegral and curvature measures
d) Steiner formula
e) Brunn-Minkowski theorem and inequality
f) Isoperimetric inequality
g) Alexandrov-Fenchel inequality
h) Strongly isomorphic polyhedra
i) Minkowski theorem on existence of a convex body with a given surface measure
If time permits we will also discuss the proof of Alexandrov's theorem on the characterisation of a sphere.
Main fields of studies for MISMaP
Type of course
Mode
Requirements
Prerequisites
Prerequisites (description)
Course coordinators
Learning outcomes
Knowledge of the basic concepts and theorems of convex geometry and Brunn-Minkowski theory.
Assessment criteria
Oral exam.
Attendance and activity in the exercises may act positively on the final grade.
Bibliography
[1] Daniel Hug, Wolfgang Weil, "Lectures on convex geometry", Springer, 2020
[2] Rolf Schneider, "Convex bodies: the Brunn-Minkowski theory", Cambridge University Press, 2014
[3] H. Federer, "Curvature measures", Trans. Amer. Math. Soc. , Vol. 93, 1959
[4] Tyrrell Rockafellar, "Convex analysis", Princeton University Press, 1970
[5] Matias Delgadino, Francesco Maggi, "Alexandrov's theorem revisited", Analysis & PDE , Vol. 12, No. 6, 2019
Additional information
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