Let $\overrightarrow{x}=(x_1,x_2, \ldots ,x_n),\overrightarrow{a}=(a_1,a_2, \ldots ,a_n)$ and $\overrightarrow{b}=(b_1,b_2, \ldots ,b_n)$ be vectors in ${\mathbb R}^n$, with $a_k \leq b_k$ for every $k$. Denote by ${\cal S}_d(\overrightarrow{x},\overrightarrow{a},\overrightarrow{b})$ the set of all polynomials $P$ of degree $\leq d$ that satisfy the interpolation constraints
$$ a_k \leq P(x_k) \leq b_k (1\leq k \leq n). $$
If $I$ is a subset of $\lbrace 1,2, \ldots n \rbrace$, say $I=\lbrace i_1 \lt i_2 \lt \ldots i_r \rbrace$, I denote by $x_I$ the vector of ${\mathbb R}^r$ defined by $\overrightarrow{x}_I=(x_{i_1},x_{i_2}, \ldots ,x_{i_r})$, and I define $\overrightarrow{a}_I$ and $\overrightarrow{b}_I$ similary.
By analogy with Helly’s theorem, one might conjecture that
$$ {\cal S}_d(\overrightarrow{x},\overrightarrow{a},\overrightarrow{b})\neq\emptyset \ \text{iff} \ {\cal S}_d(\overrightarrow{x}_I,\overrightarrow{a}_I,\overrightarrow{b}_I)\neq\emptyset \ \text{for any } I \subseteq \lbrace 1,2, \ldots n \rbrace \ \text{with} \ |I|=d+2 \tag{1} $$
Note that (1) is easily seen to be true when $d=0$ or $1$. For a polynomial $P$, denote by $c_d(P)$ the coefficient corresponding to degree $d$ in $P$. Let ${\cal C}_d(\overrightarrow{x},\overrightarrow{a},\overrightarrow{b})= c_d({\cal S}_d(\overrightarrow{x},\overrightarrow{a},\overrightarrow{b}))$ (so that ${\cal C}_d(\overrightarrow{x},\overrightarrow{a},\overrightarrow{b})$ is a possibly empty interval of $\mathbb R$). I also ask if it is true that
$$ {\cal C}_d(\overrightarrow{x},\overrightarrow{a},\overrightarrow{b})= \bigcap_{I \subseteq \lbrace 1,2, \ldots n \rbrace, |I|=d+2} {\cal C}_d(\overrightarrow{x}_I,\overrightarrow{a}_I,\overrightarrow{b}_I) \ \tag{2} $$
Again, this is easily seen to be true when $d=0$ or $1$.