Large geodesically convex subsets of tori - MathOverflow

Large geodesically convex subsets of tori

2012-01-16T21:00:23Z

Let $X=\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and let $E$ be a proper open subset of $X$. We say $E$ is geodesically convex if for any $x,y\in E$ the shortest geodesic connecting $x$ and $y$ lies in $E$.

Question. How large can the Haar/Lebesgue measure of $E$ can be?

For example, is $d=2$, then it seems that this cannot exceed $1/2$. Say, $[0,1)\times [0,s)$ is geodesically convex if and only if $s\leq1/2$. (If $s>1/2$, then $[x,x+\delta]$ is not the shortest geodesic for any $\delta\in(1/2,s)$ and any $x\in(0,1)$.)

Is it true for any $d\ge2$ that the measure of such an $E$ cannot exceed $1/2$?

Answer by Zarathustra for Large geodesically convex subsets of tori

2012-01-16T23:09:01Z

Let me write down the steps.

Consider the case d=2, the generalization is straightforward.
There is an open ball B that doesn't intersect E.
Consider two families of geodesic circles F_1 and F_2. F_1 has slope 1/p and F_2 has slope 1-1/p. Number p is chosen in such a way that each circle from F_1 nd F_2 intersects B.
Claim: $Leb(E\cap F_1(x))\le 1/2 length(F_1(x))$ OR $Leb(E\cap F_2(x))\le 1/2 length(F_1(x))$ for each x.
Proof: $E\cap F_1(x)$ is a proper union of open intervals that are separated by gaps of length at least $\sqrt{p^2+1}/2p$. So it remains to show that there are at least $p$ gaps. If there are less than $p$ gaps then there is an interval from $E\cap F_1(x)$ of length greater than $\sqrt{p^2+1}/2p$. It follows that one can find a simple closed curve C_1 in E which is C^0 close to a horizontal generator. Assume that in the same way we also can find C_2 in E which is C^0 close to the vertical generator. Then one can easily see that E is the torus and the claim follows.
Apply Fubini.

Answer by Will Jagy for Large geodesically convex subsets of tori

2012-01-16T23:25:48Z

A bit long for a comment. You can have a geodesically convex set that is an arbitrarily large proportion of the area of a surface. What I have in mind resembles a bulb_thermometer or turkey_baster but is at least $C^\infty.$ It is rotationally symmetric. One end is long, cigar shaped, half of something that approximates a prolate spheroid. It differs from an actual prolate spheroid in that it is necessary for the Gauss curvature to be 0 along the "equator," the closed geodesic where the half cigar joins the bulb. Therefore the curvature must approach 0 near the equator. Immediately upon entering the bulb section, the curvature is slightly negative, which is the reason geodesics leaving the equator cannot quickly return.

Answer by Anton Petrunin for Large geodesically convex subsets of tori

2012-01-17T06:23:27Z

(This is a new answer; my original answer was completely wrong.)

Assume $\mathop{\rm vol}E>\tfrac12$. Then it contains two opposite points say $x$ and $x'=x+(\tfrac12,\tfrac12,\dots,\tfrac12)$. WLOG we can assume that $x=0$. Taking minimizing geodesics form $(\tfrac12,\tfrac12,\dots,\tfrac12)$ to $y\approx 0$, we get that all main diagonal of unit cube $$\square^n=(0,1)\times(0,1)\times\dots\times(0,1)$$ lie in $E$. Then apply the following lemma:

Trivial Lemma. Let $\square^n$ be open unit cube in $\mathbb R^n$ and $E\subset \square^n$ be a locally convex open set which contains all main diagonals of $\square^n$ then $E=\square^n$.

To prove the lemma, note that local convexity + conectedness in $\mathbb R^n$ implies convexity.