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Sobolev space - Wikipedia

february 2017 by nhaliday

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

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february 2017 by nhaliday

Prékopa–Leindler inequality | Academically Interesting

february 2017 by nhaliday

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

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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

february 2017 by nhaliday

The Brunn-Minkowski Inequality | The n-Category Café

february 2017 by nhaliday

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).

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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).

february 2017 by nhaliday

ag.algebraic geometry - Why do combinatorial abstractions of geometric objects behave so well? - MathOverflow

q-n-a overflow math math.CO geometry synthesis intuition soft-question todo regularity math.AG math.RT polynomials positivity monotonicity nibble abstraction signum guessing

january 2017 by nhaliday

q-n-a overflow math math.CO geometry synthesis intuition soft-question todo regularity math.AG math.RT polynomials positivity monotonicity nibble abstraction signum guessing

january 2017 by nhaliday

ca.analysis and odes - Why do functions in complex analysis behave so well? (as opposed to functions in real analysis) - MathOverflow

january 2017 by nhaliday

Well, real-valued analytic functions are just as rigid as their complex-valued counterparts. The true question is why complex smooth (or complex differentiable) functions are automatically complex analytic, whilst real smooth (or real differentiable) functions need not be real analytic.

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january 2017 by nhaliday

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