Let $X$ be some convex object in the plane. For every pair of points $A,B$ that are not in $X$, and the distance between them is at most $1$, we remove from $X$ its intersection with the segment $AB$. For example, in the picture below, $X$ is the green circle, and if the distance between $A$ and $B$ is at most $1$, then the entire black segment is removed from $X$:
Note that we first find all A,B with distance at most $1$ and define the to-be-removed segments, and then remove them all at once. Denote the new object (after the removals) by $Y$.
I am trying to understand how $Y$ looks like.
If $X$ is a disc (with diameter larger than $1$), then it is easy to see that $Y$ is a disc too: it has the same center as $X$ and a smaller radius.
MY QUESTION: if $X$ is an arbitrary convex object, is $Y$ convex too?
I'll consider closed sets only, as you mentioned in your comment. Please let us denote regions with $X,Y,Z,\dots$ and reserve $A,B,C,\dots$ for points, especially let $X$ be the total region. If $[A,B]$ is a segment as described in your question, we can assume w.l.o.g. that removing it dissects $X$ into two parts $Y,Z$ (the other case is easy). My claim is that after removing all segments of your kind, either $Y$ or $Z$ vanishes. Assume not, then you have points $y\in Y,z\in Z$ which are not part of any such segment. In particular you can choose segments $[C,D]\subseteq Y$ containing $y$ and $[E,F]\subseteq Z$ containing $z$, both of length $1$ and paralell to $[A,B]$. Since $X$ is convex and $C,D,E,F\in X$, the segments $[C,E]$ and $[D,F]$ must be fully contained in $X$ as well. But also at least one of those segments must intersect with the line through $A,B$ outside of $(A,B):=[A,B]\setminus\{A,B\}$, a contradiction, as no such point is contained in $X$.
Hence for each line segment $[A,B]$ you remove, there is a set of line segments whose removal includes all points to the left or all points to the right of $[A,B]$. So removing all your line segments is equivalent to intersecting with a set of open half-planes, which yields a convex set. It can also be checked that the set is closed.