nhaliday + brunn-minkowski   6

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
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january 2017 by nhaliday

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