Let $A=(a_{ij})\in M_{m \times n}(\mathbb{R_{+}})$ and $B=(b_{ij}) \in M_{n \times l}(\mathbb{R}_{+}).$ The product of $A$ and $B$ in max algebra is denoted by $A\otimes B,$ where $(A\otimes B)_{ij}=\displaystyle\max_{k=1,\ldots,n} a_{ik}b_{kj}.$
A set $\mathcal{X}_{n \times k} \subset M_{n \times k}(\mathbb{R}_{+})$ is defined by
$$\mathcal{X}_{n \times k}= \{X \in M_{n \times k}(\mathbb{R}_{+}): X^{t}\otimes X = I_{k}\}.$$
It is known that for the case $k = n,$ $\mathcal{X}_{n \times n}$ is equal to $\mathcal{U}_{n},$ where $\mathcal{U}_{n}$ is a unitary matrix in max algebra.
I have a partial answer for this question but I am not sure if it is correct in general?
If $1 < k \leq \frac{n}{2},$ then $\mathcal{X}_{n \times k}$ is connected (in fact it's a path connected).
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