Convex bodies with constant maximal section function in odd dimensions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:42:19Z http://mathoverflow.net/feeds/question/70391 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70391/convex-bodies-with-constant-maximal-section-function-in-odd-dimensions Convex bodies with constant maximal section function in odd dimensions fedja 2011-07-15T02:08:43Z 2012-01-04T02:00:11Z <p>In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ is trivial and can be described as follows:</p> <p>Let $f:[-1,1]\to\mathbb R$ be continuous, strictly concave and satisfy $f(-1)=f(1)=0$. For every such function, let $Q_f$ be the body of revolution given by $y^2+z^2+t^2\le f(x)^2$. Then $Q_f$ and $Q_g$ have the same maximal sections in every direction if (and, actually, only if) $f$ and $g$ are equimeasurable, i.e., $|{f>t}|=|{g>t}|$ for all $t>0$. (Of course, there are plenty of concave functions equimeasurable with $\sqrt{1-x^2}$).</p> <p>With some extra work, one can construct something like this in $\mathbb R^n$ when $n$ is even though I do not know any similarly nice geometric description of such bodies for $n\ge 6$.</p> <p>What I (and my co-authors) are currently stuck with is the case of odd $n$ (say, the usual space $n=3$). In view of such simple example in $\mathbb R^4$, I suspect that we are just having a mental block. Can anybody help us out? </p> http://mathoverflow.net/questions/70391/convex-bodies-with-constant-maximal-section-function-in-odd-dimensions/84849#84849 Answer by fedja for Convex bodies with constant maximal section function in odd dimensions fedja 2012-01-04T02:00:11Z 2012-01-04T02:00:11Z <p>To those who are still interested: we've finally <a href="http://arxiv.org/PS_cache/arxiv/pdf/1201/1201.0393v1.pdf" rel="nofollow">made it</a> but it's so ugly that a nice alternative approach will be always welcome :) We are still stuck with Bonnesen's question about the possibility to recover a convex body from the volumes of its maximal sections and projections in odd dimensions, so some help will be appreciated. The even-dimensional case can be found <a href="http://arxiv.org/PS_cache/arxiv/pdf/1112/1112.3976v1.pdf" rel="nofollow">here</a>. I feel a bit like a student asking for help with his homework, of course, but why not? We all get stuck now and then :). This should really be a comment but it's too long to fit the number of characters restriction.</p>