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Reading a miniature 16 in the book "Thirty-three miniatures" by Jirí Matoušek I can understand the proof only by writing $x_\varnothing = 1$.

In that proof you observe a linear combination $\sum_{I \subseteq \{1,2,...,d\}}\alpha_Ix_I $ of multilinear monomials of the form $x_I = \prod_{i \in I}x_i$, where $I \subseteq \{1,2,...,d\}$, and $d$ is the dimension of a field over $\mathbb{R}$.

Have you ever seen the notation $x_\varnothing = 1$, and if yes - where does it come from?

jupiter_jazz
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    $x_I$ is the product $x_1 \times \ldots \times x_l$ for $I= { 1,2, \ldots , l } \subseteq { 1,2,\ldots, d }$, i.e. for $l \le d$. When $I = \emptyset$ there are no $x_i$ to multiply and the convention is adopted that the mult of no numbers is equal to $1$. – Mauro ALLEGRANZA Apr 09 '18 at 11:54

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If $x_I = \prod_{i\in I} x_i$ is a definition, then for $I=\varnothing$ you get the empty product which is defined to be $1$, since $1$ is the unit with respect to multiplication. (Just like the empty sum is $0$)

Christoph
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