Synthetic Proof for Ratio of Volumes of Concentric Spheres? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:52:24Z http://mathoverflow.net/feeds/question/41453 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41453/synthetic-proof-for-ratio-of-volumes-of-concentric-spheres Synthetic Proof for Ratio of Volumes of Concentric Spheres? Daniel Litt 2010-10-07T20:35:51Z 2010-10-07T21:17:11Z <p>Let $B^n(r)$ be the $n$-ball of radius $r$. A standard (easy) problem for first year calculus students is the following. </p> <blockquote> <p>$(1)$ Show that $$\lim_{n\to \infty} \frac{\text{Vol}(B^n(r))}{\text{Vol}(B^n(1)\setminus B^n(r))}=0$$ for $0\leq r&lt;1$. </p> </blockquote> <p>An equivalent problem is: </p> <blockquote> <p>$(2)$ Let $r_n$ be such that $2\cdot\text{Vol}(B^n(r_n))=\text{Vol}(B^n(1))$; show that $$\lim_{n\to \infty} r_n=1.$$</p> </blockquote> <p>There is a synthetic proof of (1) for $r\leq 1/2$, which goes as follows. Let $C$ be a cube of side-length $2r$ circumscribing $B^n(r)$. Then one may place $2n$ disjoint hemispheres of radius $r$ on the faces of $C$ (with the bases of the hemispheres inscribed in the faces of $C$) such that these hemispheres are all contained in $B^n(r)$. But then we must have that $$\text{Vol}(B^n(1)\setminus B^n(r))\geq n \cdot \text{Vol}(B^n(r))$$ which completes the proof. All this really uses is the Pythagorean theorem, to check that the hemispheres are contained in $B^n(1)$.</p> <p>This proof is unsatisfying for two reasons---first, it only seems to work for $r\leq 1/2$, and second, the rate of convergence one gets is $\sim1/n$, rather than the actual exponential convergence.</p> <p>So my question is</p> <blockquote> <p>Is there a synthetic proof of (1) or (2) that gives the correct convergence rate? Is there a synthetic proof of (1) for $r>1/2$?</p> </blockquote> http://mathoverflow.net/questions/41453/synthetic-proof-for-ratio-of-volumes-of-concentric-spheres/41458#41458 Answer by Gideon Schechtman for Synthetic Proof for Ratio of Volumes of Concentric Spheres? Gideon Schechtman 2010-10-07T21:17:11Z 2010-10-07T21:17:11Z <p>${\text{Vol}(B^n(r))}=r^n{\text{Vol}(B^n(1))}$ by the homogeneity of degree $n$ of the Lebegue measure. Consequently, $\frac{\text{Vol}(B^n(r))}{\text{Vol}(B^n(1)\setminus B^n(r))}=\frac{r^n}{1-r^n}\to 0$ exponentialy fast.</p> <p>But maybe this is too analytic?</p>