math.mg   52

« earlier    

inequalities - Is the Jaccard distance a distance? - MathOverflow
Steinhaus Transform
the referenced survey: http://kenclarkson.org/nn_survey/p.pdf

It's known that this transformation produces a metric from a metric. Now if you take as the base metric D the symmetric difference between two sets, what you end up with is the Jaccard distance (which actually is known by many other names as well).
q-n-a  overflow  nibble  math  acm  sublinear  metrics  metric-space  proofs  math.CO  tcstariat  arrows  reduction  measure  math.MG  similarity  multi  papers  survey  computational-geometry  cs  algorithms  pdf  positivity  msr  tidbits  intersection  curvature  convexity-curvature  intersection-connectedness  signum 
february 2017 by nhaliday
Prékopa–Leindler inequality | Academically Interesting
Consider the following statements:
1. The shape with the largest volume enclosed by a given surface area is the n-dimensional sphere.
2. A marginal or sum of log-concave distributions is log-concave.
3. Any Lipschitz function of a standard n-dimensional Gaussian distribution concentrates around its mean.
What do these all have in common? Despite being fairly non-trivial and deep results, they all can be proved in less than half of a page using the Prékopa–Leindler inequality.

ie, Brunn-Minkowski
acmtariat  clever-rats  ratty  math  acm  geometry  measure  math.MG  estimate  distribution  concentration-of-measure  smoothness  regularity  org:bleg  nibble  brunn-minkowski  curvature  convexity-curvature 
february 2017 by nhaliday
The Brunn-Minkowski Inequality | The n-Category Café
For instance, this happens in the plane when A is a horizontal line segment and B is a vertical line segment. There’s obviously no hope of getting an equation for Vol(A+B) in terms of Vol(A) and Vol(B). But this example suggests that we might be able to get an inequality, stating that Vol(A+B) is at least as big as some function of Vol(A) and Vol(B).

The Brunn-Minkowski inequality does this, but it’s really about linearized volume, Vol^{1/n}, rather than volume itself. If length is measured in metres then so is Vol^{1/n}.

...

Nice post, Tom. To readers whose background isn’t in certain areas of geometry and analysis, it’s not obvious that the Brunn–Minkowski inequality is more than a curiosity, the proof of the isoperimetric inequality notwithstanding. So let me add that Brunn–Minkowski is an absolutely vital tool in many parts of geometry, analysis, and probability theory, with extremely diverse applications. Gardner’s survey is a great place to start, but by no means exhaustive.

I’ll also add a couple remarks about regularity issues. You point out that Brunn–Minkowski holds “in the vast generality of measurable sets”, but it may not be initially obvious that this needs to be interpreted as “when A, B, and A+B are all Lebesgue measurable”, since A+B need not be measurable when A and B are (although you can modify the definition of A+B to work for arbitrary measurable A and B; this is discussed by Gardner).
mathtariat  math  estimate  exposition  geometry  math.MG  measure  links  regularity  survey  papers  org:bleg  nibble  homogeneity  brunn-minkowski  curvature  convexity-curvature 
february 2017 by nhaliday
mg.metric geometry - Pushing convex bodies together - MathOverflow
- volume of intersection of colliding, constant-velocity convex bodies is unimodal
- pf by Brunn-Minkowski inequality
q-n-a  overflow  math  oly  tidbits  geometry  math.MG  monotonicity  measure  spatial  dynamical  nibble  brunn-minkowski  intersection  curvature  convexity-curvature  intersection-connectedness 
january 2017 by nhaliday
Ehrhart polynomial - Wikipedia
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.
math  math.MG  trivia  polynomials  discrete  wiki  reference  atoms  geometry  spatial  nibble  curvature  convexity-curvature 
january 2017 by nhaliday

« earlier    

related tags

254a_-_hilbert's_fifth_problem  acm  acmtariat  additive-combo  admin  alg-combo  algebra  algorithms  amt  applications  approximate_groups  arrows  art  arxiv.org  atoms  ben-recht  ben_green  biases  binomial  blog  boltzmann  boolean-analysis  brunn-minkowski  brunn-minkowski_inequality  cartoons  characterization  classic  clever-rats  coarse-fine  combo-optimization  compressed-sensing  computational-geometry  concentration-of-measure  concept  conference  constraint-satisfaction  convexity-curvature  cool  course  cs  curiosity  current-events  curvature  data-science  differential  dimensionality  direction  discrete  distribution  duality  dynamical  embeddings  emmanuel_breuillard  entropy-like  erdos  erik-demaine  estimate  ethical-algorithms  examples  existence  expert-experience  expert  exposition  expository  extrema  fedja  freiman's_theorem  geography  geometry  giants  gleason_metrics  googlereader  gotchas  government  gowers  graph-theory  graphs  greedy  gromov's_theorem  hi-order-bits  high-dimension  hilbert's_fifth_problem  history  homogeneity  ifttt  information-theory  inner-product  insight  interdisciplinary  intersection-connectedness  intersection  interview  intuition  isotropy  lattice  law  learning-theory  lecture-notes  lectures  levers  lie_algebras  lie_groups  lifts-projections  limits  linear-algebra  linear-programming  liner-notes  links  machine-learning  magnitude  maps  matching  math.at  math.ca  math.co  math.ds  math.fa  math.gn  math.gr  math.lo  math.rt  math  mathtariat  matrix-factorization  measure  metric-space  metrics  mit  monotonicity  msr  multi  news  nibble  nilpotent_groups  noncommutative_freiman_theorem  nonstandard_analysis  norms  novelty  oly  on  one-parameter_subgroups  open-problems  optics  optimization  org:bleg  org:edu  org:inst  org:mag  org:mat  org:sci  orourke  overflow  p:*  p:someday  p:whenever  paper  papers  paradox  pdf  pigeonhole-markov  play  polisci  politics  polynomial_growth  polynomials  popsci  positivity  prekopa-leindler_inequality  preprint  princeton  probabilistic-method  probability  profile  proofs  puzzles  q-n-a  quantum-info  quantum  questions  quixotic  random-matrices  random  ratty  reduction  reference  regularity  relaxation  research  rigidity  robust  rounding  s:**  sebastien-bubeck  separation  shift  signum  similarity  smoothness  social-choice  soft-question  sparsity  spatial  stat-mech  stats  stochastic-processes  stock-flow  stories  stream  sublinear  survey  synthesis  talks  tcs  tcstariat  tensors  thurston  tidbits  tip-of-tongue  todo  topics  topology  tradeoffs  trivia  unit  updates  usa  vague  video  visual-understanding  washington  wiki  wormholes  yoga  zooming  👳 

Copy this bookmark:



description:


tags: