Answer by Roland Bacher for What's a nice argument that shows the volume of...
This fly can also be killed using the Leech-lattice as a sledgehammer:The existence of the Leech lattice (an even unimodular (integral) lattice of dimension 24 and (squared) minimum 4) implies that the...
View ArticleAnswer by orangeskid for What's a nice argument that shows the volume of the...
It's enough to check in even dimension, that is for the unit ball $B_n$ in $\mathbb{C}^n$. The map $z\mapsto \pi |z|^2$ from $\mathbb{C}$ to $[0, \infty)$ is measure preserving, and so is the map...
View ArticleAnswer by George R. for What's a nice argument that shows the volume of the...
Permit me to suggest another "intuitive" approach, which hardly uses any calculations, just some basic Arithmetic (that means: Combinatorics)Well, it is a fact that even for big numbers of n...
View ArticleAnswer by gowers for What's a nice argument that shows the volume of the unit...
I've come a bit late to this particular party, but here's another argument. This one includes most of the sphere in a suitable cone.Choose a small positive number x, to be optimized later. Then the...
View ArticleAnswer by Roland Bacher for What's a nice argument that shows the volume of...
Replacing the characteristic function of the unit ball by a suitable normal distribution with spherical symmetry when computing the volume should give approximatively the correct answer.Since...
View ArticleAnswer by Bill Johnson for What's a nice argument that shows the volume of...
Given an $n$ dimensional closed bounded convex symmetric body $E$, situate it in $R^n$ so that the Euclidean ball $B$ is the ellipsoid of maximal volume contained in $E$. In 1978 Szarek, building on...
View ArticleAnswer by Guillaume Aubrun for What's a nice argument that shows the volume...
There is a simple argument by comparing to the unit ball of $\ell_1^n$.Let $K$ be the unit ball of $\ell_1^n$, i.e. the set of points with sum of coordinates (in absolute value) bounded by $1$. Then...
View ArticleAnswer by Alink for What's a nice argument that shows the volume of the unit...
A sphere packing argument (and some kissing number construction), because having smaller n-balls is equivalent to be able to pack more of them in the unity n-cube.edit: This needed to use r=1/2, not...
View ArticleAnswer by Greg Kuperberg for What's a nice argument that shows the volume of...
A calculus-free proof that the volume $V_n$ of the unit $n$-sphere goes to 0 faster than any exponential. Equivalently, the volume $r^nV_n$ of the sphere of radius $r$ goes to 0 for every $r$. It is...
View ArticleAnswer by fedja for What's a nice argument that shows the volume of the unit...
The ultimate reason is, of course, that the typical coordinate of a point in the unit ball is of size $\frac{1}{\sqrt{n}}\ll 1$. This can be turned into a simple geometric argument (as suggested by...
View ArticleAnswer by Kevin P. Costello for What's a nice argument that shows the volume...
A variation on some of the previous arguments that gives some intuition without actually doing any calculation.Consider $B_n$, the ball in $R^n$, and $C_n$, the cube $[-\frac{1}{2}, \frac{1}{2}]^n$. We...
View ArticleAnswer by Ian Agol for What's a nice argument that shows the volume of the...
Here's a geometric argument (still with a bit of calculus). The volume of the unit $n+1$-ball may be obtained from integrating the volumes of $n$-ball cross-sections from say south pole to north pole....
View ArticleAnswer by Michael Lugo for What's a nice argument that shows the volume of...
Let $f(n)$ be the n-volume of the n-sphere. Then the natural thing to ask about is not $f(n)$, but $f(n)/2^n$; this is the ratio of the volume of the n-sphere to the volume of the n-cube in which it is...
View ArticleAnswer by Vectornaut for What's a nice argument that shows the volume of the...
Edit: As Matthias pointed out, the following argument only works for the ball with radius 1/2.To measure volume, we need to agree on a unit of volume [1]. The traditional way of doing this is to set...
View ArticleAnswer by Pete L. Clark for What's a nice argument that shows the volume of...
By a strange coincidence I found myself thinking about almost this very question last week on a walk home early one morning (yes, that's correct). I wanted however only the weaker result that the ratio...
View ArticleAnswer by Matthias for What's a nice argument that shows the volume of the...
For $r=\frac{1}{2}$, I have a geometric argument.Let $I=[-\frac{1}{2},\frac{1}{2}]$.Now $Vol(D^{n-1} \times I) = Vol(D^{n-1})$. Take an annulus $A$, say with outer radius $\frac{1}{2}$ and inner radius...
View ArticleAnswer by Gil Kalai for What's a nice argument that shows the volume of the...
Maybe the fact that most point of the sphere are very close to the equator (concentration of measure) gives some conceptual explanation.
View ArticleWhat's a nice argument that shows the volume of the unit ball in $\mathbb...
Before you close for "homework problem", please note the tags.Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about...
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