A lecture on boxing inequalities, widths, and systolic geometry presented by Alexander Nabutovsky at the Hausdorff Center for Mathematics. Explore generalizations of the classical boxing inequality for bounded domains, where for Ω⊂ℝⁿ⁺¹ and positive m∈(0,n], HCₘ(Ω)≤c(m)HCₘ(∂Ω), with HCₘ denoting the m-dimensional Hausdorff content. Discover how this result extends to higher codimensions in Banach spaces and metric spaces with linear contractibility functions. Learn about applications to systolic geometry through inequalities that provide upper bounds for the widths of M⊂B in terms of volume or Hausdorff content. The lecture covers the new inequality Wₘ₋₁ᵏ⁽ᵐ⁾(M)≤const√m·vol(Mᵐ)¹/ᵐ for closed manifolds Mᵐ⊂ℝᴺ and its implications to systolic geometry, representing joint work with Sergey Avvakumov.