1
$\begingroup$

Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$.

Define linear functions $f(x)= a_1x_1+ \cdots + a_nx_n$ on $L$, of course also on $R^n$, such that $a_i > 0, i =0,\cdots,n$.

Suppose we have $L_1,\cdots,L_k$ which are disjoint subsets of L. Say $\{L_i\}_{i=1}^k$ is well-separated by $f$ if there exists $0\leq\theta_0<\theta_1<\cdots<\theta_{k-1}<\theta_{k}\leq\infty$ such that $L_i \subset \{\theta_{i-1}<f<\theta_{i}\}$ for $i=1,\cdots,k$.

I want to know what are the conditions for a fixed collection of disjoint subsets of $L$, denote as $\{L_i\}_{i=1}^k$, to be well-separated by some linear function $f$ defined as above.

For example, when $k=2$, it is easy to see well-separateness by some linear function of above is equivalent to the convex hull of $L_1$ and $L_2$ in $R^n$ are disjoint.

Now the question is whether I can have the similar condition for $k>2$.

Thanks a lot.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.