# Property of compact convex sets of width 1

## Homework Statement

A strip of width w is a part of the plane bounded by two parallel lines at distance w. The width of a set ##X \subseteq \mathbb{R}^2## is the smallest width of a strip containing ##X##. Prove that a compact convex set of width ##1## contains a segment of length ##1## in every direction.

## The Attempt at a Solution

Let ##K## be a compact convex set in the plane. Suppose for contradiction that there exists a direction, call it ##D##, for which there is no segment of length ##1## in ##K##. This means that the length of all segments contained in ##K## in this direction are between ##0## and ##1##. Let ##v## be the unit vector in direction ##D##, and consider the translate ##v+K##. ##K## and ##v+K## are disjoint, since if they weren't we could find a segment of length ##1## in the direction of ##D##. ##K## and ##v+K## are also compact and convex, so we can use the strong separating theorem to find a line that separates them, and in particular a line that is the perpendicular of the two figures and has normal vector ##v##. Note that for essentially the same reasons we can find a separating line between ##K## and ##-v+K## with the same properties.

Now. let ##\epsilon## be the orthogonal distance between the separating line of ##K## and ##v+K##, and either ##K## or ##v+K## (the distance is the same by symmetry). For simplicity sake we will use the term "right" to mean in the direction of ##v##, and the term "left" to mean in the direction of ##-v##. Now, form a line segment from the leftmost point in ##K## to the leftmost point in ##v+K##. This line segment must be of length ##1##. Now, translate this line to the left by ##\epsilon##. By symmetry, we have formed a line segment of length ##1## between the two separating lines, which means that ##K## is at most width ##1##. However, remember that our separating lines are strict, meaning that if we translate the rightmost separating line to the left by ##\epsilon##, and translate the leftmost separating line to the right by ##\epsilon##, this shows that that the width of ##K## is at most ##1-2\epsilon##, which contradicts the assumption that ##K## had width ##1##.

I have two questions. Does this proof seem to work? And if it does, is there any wordiness I could cut down on?

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