I currently have a question about the definition of convex body. The formal definition is: a convex body is a convex set which has non-empty interior. By non-empty interior, we meant for a set $N\subset M$, a point $x\in N$, and any points $y\in M$, $\exists\ \epsilon=\epsilon(y)$ such that $x + ty \in N$ for $|t|< \epsilon$. I think the definition should state that $|t|>0$ as well.
The reason I asked this question is because I was trying to prove that the Hilbert Cube is a convex set but not convex body. After proving it a convex set, and using contradiction to prove the latter part, I ended up with the case that if the Hilbert Cube is a convex body, then $t=0$ is the only value that may satisfy $x + ty\in$ Hilbert Cube. This is immediately a contradiction to the "new" definition I had above, since $|t|> 0$. I wonder if this is a valid conclusion to make?
This is not a complete statement without quantifiers on $x, y$. Did you mean: For all $x,y$? For some $x,y$? For all $x$ and some $y$ dependent on $x$? Etc.
Here's a precise statement you can prove: there is $y$ such that for every $x\in C$ ($C$ being the Hilbert cube) $x+ty\notin C$ for $t\ne 0$. (This implies that $x$ is not an interior point.) I'll give you a hint: $y$ can be chosen to be the sequence $y_k=1/k^{2/3}$.
This is an odd way to talk about nonempty interior. Maybe you confused the definition with a consequence of having nonempty interior? Normally, $x$ is an interior point if some open ball centered at $x$ is contained in the set.