This question has be stumped completely - it is a question in C. Pugh's, Real Analysis. Let me go over some definitions from within the text I'm referring to, and note that these definitions pertain to Euclidean space (where I dropped the boldface vector notation for simplicity - moreover, the book I'm using does this too, so this is for conventional purposes on my part also). This is a homework problem, and up front I thought to just answer the question flat out, but then, in class the other day, the professor mentioned that he wants us to prove our claim(s) on this problem.
Definition: A unit ball in $\mathbb{R}^{m}$ is defined as $B^{m}=\big\{x\in\mathbb{R}^{m}:|x|\leq 1\big\}\subset\mathbb{R}^{m}$. The unit sphere is defined as $S^{m-1}=\big\{x\in\mathbb{R}^{m}:|x|=1\big\}$.
Definition: A box in $\mathbb{R}^{m}$ is the Cartesian product of intervals $[a_{1},b_{1}]\times[a_{2},b_{2}]\times\cdots\times[a_{m},b_{m}]$. The unit cube in $\mathbb{R}^{m}$ is then defined as the box $[0,1]^{m}=[0,1]\times[0,1]\times\cdots\times[0,1]$.
Definition: A set $E\subset\mathbb{R}^{m}$ is convex if for each pair of point $x,y\in E$ implies the straight line segment between $x$ and $y$ is also contained in $E$.
Remark: The straight line determined by distinct point $x,y\in\mathbb{R}^{m}$ is the set of all linear combinations $sx+ty$ where $s+t=1$ and the line segment is the set of all such linear combinations where $s,t\leq 1$.
Definition: A convex combination is a linear combination $sx+ty$ with $s+t=1$ and $0\leq s,t\leq 1$.
These are all of the definitions given in the text after recapping vector addition, subtraction, etc., the dot product, the Triangle Inequality, The Euclidean distance between vectors, the Cauchy Schwarz Inequality, and the Triangle Inequality for distance all in $\mathbb{R}^{m}$.
Preliminary Notes/Work: Now that you are on the same page as me, here's why I'm asking this problem. The book poses these questions, but the professor wants us to prove these results after making our respective claim(s). I started the problem by letting $[-a,a]^{m}$ for some $a\in\mathbb{R}^{+}$ be the given cube, and then I figured to claim that the ball $B_{0}^{m}:=\big\{x\in\mathbb{R}^{m}:|x|\leq a\big\}$ is the largest ball contained in the cube (centered at the origin). From here, I figured to show $B^{m}\subset[-a,a]^{m}$ and then show that this ball is the largest by contradiction and/or to use the following:
Definition Let $S$ be a set with some property. A subset $H$ of $S$ is the largest with respect to the property if $H$ has the property and for any subset $K\subset S$ that has the property implies that $K\subset H$.
I also determined that both sets, the ball and the cube, are convex. And I started thinking that maybe the problem could be proved by showing that there exists a line segment in any ball with a radius larger than the one specified above does not lie in the given cube. Any thoughts, recommendations, suggestions, etc. will be GREATLY appreciated! Am I on the right track? Is there another way to proceed? The problem is not marked as a difficult problem in the book, so I'm thinking there must be a way to do this without too much complexity? Keep in mind that we have not yet discussed topology, other than the definition of an arbitrary topology on a set. But I have some experience with metric spaces and defining a topology on them from independent study in the past.
Hint: as discussed in the comments, we need to take "largest" to be measured by the edge length of the cubes or the radius of the balls, or equivalently by areas/volumes.
For cubes inside balls, look at the diagonal going from one corner to the opposite corner, e.g., from $(-1, -1, -1)$ to $(1, 1, 1)$ in the cube $[-1, 1]^3 \subseteq \mathbb{R}^3$. This is the longest straight line segment contained in the cube and it will have to fit inside the containing ball. (So the solution will generalise the obvious way of inscribing a large square inside a circle.)
For balls inside cubes, note that the ball must meet a face of the cube and that then the diameter of the ball at the point of intersection must be orthogonal to the face. (So the solution will generalise the obvious way of inscribing a large circle inside a square.)