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The following is an exercise from Lectures on Discrete Geometry by J. Matousek, which I find hard.

Let $C\subset \mathbb{R}^d$ be a compact convex set with a nonempty interior, and let $p\in C$ be an interior point. Show that there exists a line $\ell$ passing through $p$ such that the segment $\ell\cap C$ is at least as long as any segment parallel to $\ell$ and contained in $C$.

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It suffices to find $\ell$ such that at its endpoints $a,b$ the set $C$ admits supporting hyperplanes that are parallel to each other. I imagine that the Borsuk-Ulam theorem will be used. –  user53153 Feb 10 '13 at 6:19
I understand the "it suffices" part. But it is still puzzling to me how to deduce the conclusion using the theorem aforementioned. Could anyone give more details? @5PM –  Robin Feb 10 '13 at 14:31
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1 Answer

You can use the following statement: There is no unit vector field $v$ on $S^n$ that is anti-symmetric with respect to the central symmetry of $S^n$.

This theorem holds because if you would have such a unit field, then you would have a homothopy from $S^n$ to $S^n$ that connects the identity map with a map from $S^n$ to itself that factors through a map to $RP^n$. But any such map $S^n\to RP^n\to S^n$ has and even degree (i.e. the degree can not be $1$).

Now if you accept the statement, you can reason as follows.

Denote by $C_{\varepsilon}$ the set of points on distance at most $\varepsilon$ from $C$. It is sufficient to prove the statement for each $C_{\varepsilon}$ and then take the limit $\varepsilon\to 0$. Now, $C_{\varepsilon}$ has $C^1$-smooth boundary, i.e. at each point of $\partial C_{\varepsilon}$ there is a unique supporting hyperplane.

Suppose that for every segment $[x,y]$ that passes through $p$ and such that $x,y\in \partial C$ the supporting hyperplanes $P_x$ and $P_y$ intersect. Then chose unit vectors in $P_x$ and $P_y$ at $x$ and $y$ correspondingly, that point exactly towards $P_x\cap P_y$. This will give you the desired vector-filed (anti-symmetric with respect to the involution of $\partial C$ that exchanges the ends of segments passing through $p$).

I never saw the book so I don't know if this is the solution that is expected in this book

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Thanks. I guess that a proof without relating to homotopy and vector field is expected. The problem is an exercise just after the concept of convexity and affinity, the Carathéodory theorem and the separation theorem are presented in the book. –  Robin Feb 14 '13 at 7:57
I see, I would be curious to see a more elementary proof. –  agleaner Feb 14 '13 at 12:14
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