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I am trying to work out whether level sets of linear combinations of Gaussian functions are unique.

For a given integer $n\ge 1$, fix $n$ points $x_i\in\mathbb{R}^d$ and $\sigma>0$. Let $\mathcal{H}$ be the space of functions $f:\mathbb{R}^d\rightarrow\mathbb{R}$ that can be written $$f(x) = \sum^n_1 a_i .\text{exp}\left({-\frac{||x-x_i||^2}{\sigma^2}}\right),$$ where $a_i\in\mathbb{R}$. If we define the set $S_f$ by $$S_f:=\left\{ x\in\mathbb{R}^d: f(x)\ge 1 \right\},$$ then is it true that for $S_f$ with non-empty interior, that $S_f\ne S_{f'}$ whenever $f\ne f'$?

I know this is not true for $d=1$, but I think it should be true for $d>1$. I would be grateful if anyone had insight into this question.

Note: Of course, the constant 1 in the definition of $S_f$ above is arbitrary, and any non-zero constant would do.

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    $\begingroup$ In other words, you are asking if the vectors $(\exp\{-{|x-x_i|^2/\sigma^2}\})_{i=1}^n$ span the whole $\mathbb R^n$ when $x$ runs over the boundary of some reasonably decent bounded open set. This seems very plausible though I need to think more to find a formal proof. $\endgroup$
    – fedja
    Commented Oct 22, 2013 at 14:06
  • $\begingroup$ Yes, you are right - this would be an equivalent (and more concise) way to pose the problem. $\endgroup$
    – KNW
    Commented Oct 22, 2013 at 14:45
  • $\begingroup$ Do you insist on the same variance throughout? Meaning, in your formula for $f(x)$, should that $\sigma^2$ be a $\sigma_i^2$? $\endgroup$
    – Vidit Nanda
    Commented Oct 22, 2013 at 18:19
  • $\begingroup$ The tags 'linear-algebra' and 'kernels' are unrelated to the subject of this question. Unfortunately, I have no idea how to tag this question. Perhaps someone more knowledgeable can retag this question with more appropriate tags? Thanks. $\endgroup$
    – Ricardo Andrade
    Commented Oct 22, 2013 at 19:35
  • $\begingroup$ This problem is related to the study of Gaussian kernels (hence the tag), and in this context I have constant variance throughout. I haven't considered this question with different variances. $\endgroup$
    – KNW
    Commented Oct 23, 2013 at 9:09

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