satisfies the following conditions:

-   $h_0=h_d=1$ and $h_i=h_{d-i}$ for $i=1,2,\dots,d-1$.

-   $h_i \geq 0$ for $i=0,1,\dots,d$.

-   If $h_n=0$ for some $1 \leq n \leq d-1$ then $\sum_{k=0}^d h_k$ is even.

For a vector ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$, we define $\partial {\mathbf{h}}=(\partial h_0,\partial h_1,\dots,\partial h_{d-1}) \in { \ensuremath{\mathbb{Z}}}^d$ by $$\partial h_i = (h_0+h_1+\cdots +h_i)-(h_d+h_{d-1}+\cdots +h_{d-i})$$ for $i=0,1,\dots,d-1$. It is known that if ${\mathbf{h}}$ is the $h$-vector of a simplicial cell $(d-1)$-ball $P$, then $\partial {\mathbf{h}}$ is the $h$-vector of the boundary sphere of $P$. In this paper, we prove the next result, which characterizes all possible $h$-vectors of simplicial cell balls.

\[1.2\] Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$. Then ${\mathbf{h}}$ is the $h$-vector of a simplicial cell $(d-1)$-ball if and only if it satisfies the following conditions:

-   $h_0=1,$ $h_d=0$ and $h_k \geq 0$ for $k=1,2,\dots,d-1$.

-   $\partial h_k \geq 0$ for $k=0,1,\dots,d-1$.

-   If $d$ is odd and $\partial h_n=0$ for some $1 \leq n \leq d-2$ then $\sum_{k=0}^d h_k$ is even.

-   If $\partial h_n=0$ for some $1 \leq n \leq d-2$ then $$h_k + h_{k-1} + \cdots +h_{k-n+1} \geq \partial h_k
    \ \ \mbox{ for } k=n,n+1,\dots,d-1.$$

-   If $\partial h_i=0$ and $h_j=0$ for some positive integers $i$ 