Dan Klain's Publications

• What’s for Dessert? [full text]
  with Tanya Khovanova,
  The Mathematical Intelligencer, published online 05 August 2024 (print version to appear).
  
  
• Tetrahedra with congruent face pairs [full text] [alternate]
  The Mathematical Intelligencer, 45 (2023), no. 3, 251-255.
  
  
• A probabilistic proof of the spherical excess formula [full text] [alternate]
  American Mathematical Monthly, 128 (2021), no. 1, 70-72.
  
  
• Volume bounds for shadow covering [full text] [alternate]
  with Christina Chen and Tanya Khovanova,
  Transactions of the American Mathematical Society, 366 (2014), no. 3, 1161-1177.
  
  
• Steiner symmetrization using a finite set of directions [full text] [alternate]
  Advances in Applied Mathematics, 48 (2012), no. 2, 340-353.
  
  
• A countable set of directions is sufficient for Steiner symmetrization [full text] [alternate]
  with Gabriele Bianchi, Erwin Lutwak, Deane Yang, and Gaoyong Zhang,
  Advances in Applied Mathematics, 47 (2011), no. 4, 869-873.
  
  
• On the equality conditions of the Brunn-Minkowski theorem [full text] [alternate]
  Proceedings of the American Mathematical Society, 139 (2011), no. 10, 3719-3726.
  
  
• If you can hide behind it, can you hide inside it? [full text] [alternate]
  Transactions of the American Mathematical Society, 363 (2011), no. 9, 4585-4601.
  
  
• Containment and inscribed simplices [full text] [alternate]
  Indiana University Mathematics Journal, 59 (2010), no. 4, 1231-1244.
  
  
• Covering shadows with a smaller volume [full text] [alternate]
  Advances in Mathematics, 224 (2010), no. 2, 601-619.
  
  
• Angles as probabilities [full text] [alternate]
  with David V. Feldman,
  American Mathematical Monthly, 116 (2009), no. 8, 732-735.
  
  
• Estimation of Euler characteristic from volumetric data [proceedings full text (pdf)]
  with K. Rybnikov, K. Daniels, B. Jones, and V. Durante,
  Proceedings of the Fourth International Symposium on 3D Data Processing,
  Visualization, and Transmission, Georgia Institute of Technology, June 18-20, 2008.
  
  
• Bonnesen-type inequalities for surfaces of constant curvature [full text] [alternate]
  Advances in Applied Mathematics, 39 (2007), no. 2, 143-154.
  
  
• Isometry invariant valuations on hyperbolic space [full text] [alternate]
  Discrete and Computational Geometry, 36 (2006), no. 3, 457-477.
  
  
• An error estimate for the isoperimetric deficit [full text] [alternate]
  Illinois Journal of Mathematics, 49 (2005), no. 3, 981-992.
  
  
• An intuitive derivation of Heron's formula [full text] [alternate]
  American Mathematical Monthly, 111 (2004), no. 8, 709-712.
  
  
• The Minkowski problem for polytopes [abstract] [full text]
  Advances in Mathematics, 185 (2004), no. 2, 270–288.
  
  
• Continuous and profinite combinatorics [full text]
  with Joseph P. S. Kung,
  in Gian-Carlo Rota on Analysis and Probability,
  edited by Jean Dhombres, Joseph Kung and Norton Starr.
  (Boston: Birkhäuser Verlag, 2003).
  
  
• Free polygon enumeration and the area of an integral polygon [abstract] [full text]
  Discrete Mathematics, 218 (2000), 109-119.
  
  
• Even valuations on convex bodies [abstract] [full text]
  Transactions of the American Mathematical Society, 352 (2000), no. 1, 71-93.
  
  
• An Euler relation for valuations on polytopes [abstract] [full text]
  Advances in Mathematics, 147 (1999), no. 1, 1-34.
  
  
• Kinematic formulas for finite vector spaces [abstract] [full text]
  Discrete Mathematics, 179 (1998), 121-132.
  
  
• Kinematic formulas for finite lattices [abstract] [full text]
  Annals of Combinatorics, 1 (1997), no. 4, 353-366.
  
  
• A continuous analogue of Sperner's theorem [abstract] [full text]
  with Gian-Carlo Rota,
  Communications on Pure and Applied Mathematics, 50 (1997), no. 3, 205-223.
  
  
• Invariant valuations on star-shaped sets [abstract] [full text]
  Advances in Mathematics, 125 (1997), no. 1, 95-113.
  
  
• Star valuations and dual mixed volumes [abstract] [full text]
  Advances in Mathematics, 121 (1996), no. 1, 80-101.
  
  
• A short proof of Hadwiger's characterization theorem [abstract] [full text]
  Mathematika, 42 (1995), no. 2, 329-339.
  
  
• Several of my preprints are also available at arXiv.org

• Introduction to Geometric Probability, with Gian-Carlo Rota.
  (New York: Cambridge University Press, 1997).
  

• Essentials of Number Theory, draft preprint version, 2019.
  
  

• Dehn-Sommerville relations for triangulated manifolds [pdf]
  
  
• The Minkowski problem for polygons [pdf]
  
  
• The Minkowski problem for simplices [pdf]
  
  
• The Minkowski condition and divergence [pdf]
  
  
• Lecture notes on the simplex method [pdf]
  
  
• Lecture notes on matrix exponentials [pdf]
  
  
• Lecture notes on orthogonal matrices [pdf]
  
  
• Lecture notes on projection matrices [pdf]
  
  
• Lecture notes on rotation matrices [pdf]