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\newcommand{\matt}[1]{{\color{blue} \sf  Matt: [#1]}}
\newcommand{\jeronimo}[1]{{\color{red} \sf  Jer\'onimo: [#1]}}
\newcommand{\danai}[1]{{\color{cyan} \sf  Danai: [#1]}}

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\title[Inequalities for $f^*$-vectors of Lattice Polytopes]{Inequalities for $f^*$-vectors of Lattice Polytopes}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Matthias Beck, Danai Deligeorgaki, Max Hlavacek \and Jer\'onimo Valencia-Porras]{Matthias Beck\thanks{\href{mailto:becksfsu@gmail.com}{becksfsu@gmail.com}.}\addressmark{1}, Danai Deligeorgaki\thanks{\href{mailto:danaide@kth.se}{danaide@kth.se}. Danai Deligeorgaki was supported by
the Knut and Alice Wallenberg Foundation.}\addressmark{2},  \\ Max Hlavacek\thanks{\href{mhlava@math.berkeley.edu}{mhlava@math.berkeley.edu}}\addressmark{3}, \and Jer\'onimo Valencia-Porras\thanks{\href{j2valenc@uwaterloo.ca}{j2valenc@uwaterloo.ca}}\addressmark{4}}

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\address{\addressmark{1}Department of Mathematics,
         San Francisco State University,
         San Francisco, CA 94132,
         USA \\ \addressmark{2}Institutionen f\"or Matematik, KTH, SE10044, Stockholm, Sweden
          \\ \addressmark{3}Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA
           \\ \addressmark{4}Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{
    The Ehrhart polynomial $\text{ehr}_P(n)$ of a lattice polytope $P$ counts the number
of integer points in the $n$-th dilate of $P$. The $f^*$-vector of $P$,
introduced by Felix Breuer in 2012, is the vector of coefficients of $\text{ehr}_P(n)$
with respect to the binomial coefficient basis  $
\left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d = \text{dim} P$.
   Similarly to $h/h^*$-vectors, the $f^*$-vector of $P$ coincides with the $f$-vector of
its unimodular triangulations (if they exist). 
   We present several inequalities that hold among the coefficients of $f^*$-vectors of polytopes.
   These inequalities resemble striking similarities with existing inequalities for the
coefficients of $f$-vectors of simplicial polytopes; e.g., the first half of the
$f^*$-coefficients increases and the last quarter decreases. 
   Even though $f^*$-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. 
   We also show that for any polytope with a given Ehrhart $h^*$-vector, there is a polytope with the same $h^*$-vector whose $f^*$-vector is unimodal.}


%\resume{\foreignlanguage{greek}{Για ένα πολύτοπο $P$ διάστασης $d$ του οποίου οι κορυφές έχουν ακέραιες συντεταγμένες, το πολυώνυμο \foreignlanguage{english}{Ehrhart} $\text{\foreignlanguage{english}{ehr}}_P(n)$ απαριθμεί το πλήθος ακέραιων σημείων της $n$-οστής διόγκωσης του $P$. Το διάνυσμα $f^*(P)$, που μελετήθηκε αρχικά από τον \foreignlanguage{english}{Felix Breuer} το 2012, είναι το διάνυσμα συντελεστών του $\text{\foreignlanguage{english}{ehr}}_P(n)$, ως προς τη βάση διωνυμικών συντελεστών $\left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$. Όμοια με την περίπτωση των διανυσμάτων $h(P) $ και $h^*(T) $, το διάνυσμα $f^*(P)$ συμπίπτει με το διάνυσμα $f(T)$ για κάθε μονομετρικό τριγωνισμό $T$ του $P$ (εφόσον υπάρχει). Σε αυτό το άρθρο παρουσιάζονται σχέσεις ανισότητος μεταξύ των συντελεστών του διανύσματος $f^*(P)$. Αυτές οι ανίσοτητες έχουν εντυπωσιακή ομοιότητα με ανάλογες ανισότητες που ισχύουν για $f$-διάνυσματα μονοπλεκτικών πολυτόπων· παραδείγματος χάριν, οι συντελεστές του $f^*(P)$ αυξάνονται κατά το πρώτο μισό του διανύσματος και φθίνουν κατά το τελευταίο τέταρτο. Επιπλέον, σε αυτό το άρθρο αποδεικνύεται ότι, παρόλο που το διάνυσμα $f^*(P)$ δεν είναι απαραιτήτως μονότροπο, υπάρχουν αρκετές οικογένειες πολύτοπων που έχουν αυτή την ιδιότητα. Επίσης δείχνουμε ότι, δεδομένου του $h^*$-διανύσματος ενός πολυτόπου, υπάρχει πολύτοπο με το ίδιο $h^*$-διάνυσμα του οποίου το $f^*$-διάνυσμα είναι μονότροπο.}}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Lattice polytope, Ehrhart polynomial, Gorenstein polytope, $f^*$-vector, $h^*$-vector, unimodality.}

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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above
\section{Introduction}\label{intro}
For a $d$-dimensional \emph{lattice polytope} $P \subset \R^{d}$ (i.e., the convex hull of finitely many points in $\Z^d$) and a positive integer $n$, let $\ehr_P(n)$ denote the number of integer lattice points in $nP$. 
Ehrhart's famous theorem~\cite{ehrhartpolynomial} says that $\ehr_P(n)$ evaluates to a
polynomial in $n$.
Similar to the situations with other combinatorial polynomials, it is useful to express
$\ehr_P(n)$ in different bases; here we consider two such bases consisting of binomial
coefficients:
\begin{equation}\label{eq1}
  \ehr_P(n)
  \ = \ \sum_{k=0}^d h^*_k \binom{n+d-k}{d}
  \ = \ \sum_{k=0}^d f^*_k \binom{n-1}{k} \, .
\end{equation}
We call $(f^*_0, f^*_1, \dots, f^*_d)$ the \emph{$f^*$-vector} and 
$(h^*_0, h^*_1, \dots, h^*_d)$ the \emph{$h^*$-vector} of~$P$.
Stanley~\cite{stanleymagiccohenmac} proved that the $\hstar$-vector of any lattice polytope
is nonnegative (whereas the coefficients of $\ehr_P(n)$ written in the standard monomial basis can be negative).
Breuer~\cite{breuerehrhartf} proved that the $f^*$-vector of any lattice polytopal complex is
nonnegative (whereas the $\hstar$-vector of a complex
can have negative coefficients); his motivation was that various combinatorially-defined polynomials can be
realized as Ehrhart polynomials of complexes and so the nonnegativity of the $f^*$-vector
yields a strong constraint for these polynomials.
The $f^*$- and $h^*$-vector can also be defined through the \emph{Ehrhart series} of $P$:
\[
  \Ehr_P(z)
  \ := \ 1 + \sum_{n\geq1} \ehr_P(n) \, z^n
  \ = \ \frac{ \sum_{ k=0 }^d h^*_k \, z^k }{(1-z)^{d+1}} 
  \ = \ 1 + \sum_{ k=0 }^d f^*_k \left( \frac{ z }{ 1-z } \right)^{k+1} .
\]
It is thus sometimes useful to add the definition $f_{-1}^* := 1$.
The polynomial $\sum_{ k=0 }^d h^*_k \, z^k$ is the \emph{$h^*$-polynomial} of $P$, and
its degree is the \emph{degree} of~$P$. For general background on Ehrhart theory, see,
e.g.,~\cite{ccd}.
The $f^*$- and $h^*$-vectors share the same relation as $f$- and $h$-vectors of 
polytopes/polyhe\-dral complexes, namely
\begin{equation}\label{eq2}
    \sum_{k=0}^d h_k^* \, z^k \ = \ \sum_{k=0}^{d+1}f_{k-1}^* \, z^{k}(1-z)^{d-k+1}
\end{equation}
\begin{equation}{\label{h* and f*}}
    h_k^* \ = \ \sum_{j=-1}^{k-1} (-1)^{k-j-1}\binom{d-j}{k-j-1}f_j^* 
\end{equation}
\begin{equation}{\label{f* and h*}}
    f_k^* \ = \ \sum_{j=0}^{k+1} \binom{d-j+1}{k-j+1}h_j^* \, .
\end{equation}
The (very special) case that $P$ admits a unimodular triangulation yields the strongest
connection between $f^*$/$h^*$-vectors and $f$/$h$-vectors: in this case the
$f^*$/$h^*$-vector of $P$ equals the $f$/$h$-vector of the triangulation, respectively.
\begin{example}\label{example with cube}
Let $P$ be the $2$-dimensional cube $[-1,1]^2$.
The unimodular triangulation of $P$ shown in \cref{fig: small cube}, has $f$-vector
$(f_0,f_1,f_2)=(9,16,8)$, as $f_i$ counts its $i$-dimensional faces. Equivalently,
\[f^*(P)=(9,16,8) \, , \]
and one easily checks that \eqref{eq1} yields the familiar Ehrhart polynomial $\ehr_P(n) =
(2n+1)^2$.
\begin{figure}
\centering
  \begin{tikzpicture}[-
  ,node distance=1.5cm,
            main node/.style={circle, blue,fill,  inner sep=3pt}]
\centering
  \node[main node] (2) { };
  \node[main node] (1) [left of=2] { };
  \node[main node] (3) [ right of=2] { };
    \node[main node] (4) [ above of=3] { };
    \node[main node] (5) [ left of=4] { };
     \node[main node] (6) [ left of=5] { };
     \node[main node] (7) [ above of=6] { };
     \node[main node] (8) [ right of=7] { };
     \node[main node] (9) [ right of=8] { };
   
  \path[every node/.style={font=\sffamily\small}]
    [color=blue] (2) edge node { } (1)
    (3) edge node { } (2)
        (2) edge node { } (5)
            (6) edge node { } (1)
                (3) edge node { } (4)
        (6) edge node { } (5)
            (5) edge node { } (4)
                (6) edge node { } (7)
                    (5) edge node { } (8)
                        (4) edge node { } (9)
        (9) edge node { } (8)
            (7) edge node { } (8)
                (5) edge node { } (1)
                    (5) edge node { } (9)
                        (2) edge node { } (4)
                            (6) edge node { } (8);
    \end{tikzpicture}
\caption{A (regular) unimodular triangulation of the cube $[-1,1]^2$.}\label{fig: small cube}
\end{figure}
\end{example}
\begin{example}\label{eg}
The $f^*$-vector of a $d$-dimensional unimodular simplex $\Delta$ equals
\[ \left[ \binom{d+1}{1},\binom{d+1}{2}, \dots, \binom{d+1} {d+1} \right],\]
coinciding with the $f$-vector of $\Delta$ considered as a simplicial complex.
If we append this vector by $f_{-1}^* = 1$, 
it gives the only instance of a \emph{symmetric} $f^*$-vector of a lattice polytope $P$, since the
equality $f_{-1}^*=f^*_d$ implies that $h_i^*=0$ for all $1\leq i \leq d$. 
\end{example}
    
There has been much research on (typically linear) constraints for the $h^*$-vector of a
given lattice polytope (see, e.g., \cite{stapledondelta,stapledonadditive}). On the other
hand, $f^*$-vectors seem to be much less studied, and our goal is to rectify that situation. Our motivating question is how close the $f^*$-vector of a
given lattice polytope is to being \emph{unimodal}, i.e., the $f^*$-coefficients increase
up to some point and then decrease.
Our main results are as follows.
\begin{theorem}\label{thm:main}
Let $d \geq 2$ and let $P$ be a $d$-dimensional lattice polytope. Then
\begin{enumerate}[label={\rm(\alph*)}]
\item\label{f_star_unimod_triang_first_half}
   $f^*_0<f^*_1<\cdots<f^*_{\left\lfloor \frac{d}{2}\right\rfloor-1}\leq f^*_{\left\lfloor
\frac{d}{2}\right\rfloor}$;   
\vspace{5pt}
\item\label{f_star_unimod_triang_last_quarter}
  $f^*_{ \left\lfloor \frac{3d}{4} \right\rfloor } > f^*_{ \left\lfloor \frac{3d}{4}
\right\rfloor + 1 } > \dots > f^*_d$; 
\vspace{5pt}
\item\label{f_star_0ismin}
    $f^*_k \leq f^*_{d-1-k}$ for $ 0 \leq  k \leq  \frac{(d-3)}{2}$.
\end{enumerate}
\end{theorem}
\Cref{example with cube,eg} yield cases of polytopes for which the inequalities $ f^*_{ \lfloor \frac{3d}{4} \rfloor -1 }< f^*_{ \lfloor \frac{3d}{4} \rfloor  }$ and $f^*_{\lfloor \frac{d}{2}\rfloor}>f^*_{\lfloor \frac{d}{2}\rfloor+1}$
hold, respectively, and thus the strings of inequalities in
\eqref{f_star_unimod_triang_first_half} and \eqref{f_star_unimod_triang_last_quarter} can,
in general, not be extended further.
We record the following immediate consequence of \cref{thm:main}.
\begin{corollary}\label{cor:endpointslowest}
Let $P$ be a $d$-dimensional lattice polytope. Then for $0\leq k \leq d$,
\[
  f^*_{k} \ \geq \ \min \{ f^*_{0}, f^*_{d} \} \, .
\]
\end{corollary}
We remark that one can prove that if $P$ is of degree $d\geq 2$, equal to its dimension, then
$f_0^* \le f_d^*$, except when the $h^*$-vector of $P$ has entries $h_2^*=\cdots=h_d^*=1$.

\begin{theorem}\label{thm:lowdimunimodal}
The $f^*$-vector of a $d$-dimensional lattice polytope, where $1\leq d \leq 13$, is
unimodal. On the other hand, there exists a $15$-dimensional lattice simplex with nonunimodal
$f^*$-vector.
\end{theorem}
Even though $f^*$-vectors are quite different from $f$-vectors of polytopes, the above
results resemble striking similarities with existing theorems on $f$-vectors.
Namely, Bj\"or\-ner~\cite{bjornerunimodalityconj,bjornerfacenumbers,bjornerpartialunimodal} 
proved that the $f$-vector of a simplicial
$d$-polytope satisfies all inequalities in
\cref{thm:main} 
 (with the $^*$s removed, and the last coordinate dropped). 
 In fact, Bj\"orner also showed that in the $f$-analogue of \cref{thm:main}\eqref{f_star_unimod_triang_last_quarter} the decrease starts from $\lfloor{\frac{3(d-1)}{4}}\rfloor-1$ instead of $\lfloor \frac{3d}{4} \rfloor$, and that the inequalities in \cref{thm:main}\eqref{f_star_unimod_triang_first_half} and
\eqref{f_star_unimod_triang_last_quarter} cannot be further extended, by constructing a simplicial polytope with $f$-vector that
peaks at $f_j$, for any $\lfloor \frac d 2 \rfloor \le j \le \lfloor \frac{ 3(d-1) }{ 4 }-1
\rfloor$. 
\cref{cor:endpointslowest} compares the entries of the $f^*$-vector with the minimum between the first and the last entry. Note that a similar relation for $f$-vectors of polytopes was recently proven by
Hinman~\cite{hinman}, answering a question of B\'ar\'any from the 1990s. (Hinman also
proved a stronger result, namely certain lower bounds for the ratios $\frac{ f_k }{ f_0
}$ and $\frac{ f_k }{ f_{d-1} }$.)
The $f$-analogue of \cref{thm:lowdimunimodal} is again older:
Bj\"orner~\cite{bjornerunimodalityconj} showed that the $f$-vector of any
simplicial $d$-polytope is unimodal for $d \le 15$ (later improved to $d\leq19$ by Eckhoff~\cite{eckhoffdimension}), and he and Lee~\cite{billeralee}
produced examples of 20-dimensional simplicial polytopes with nonunimodal $f$-vectors.
For a special class of polytopes we can increase the range in
\cref{thm:main}\eqref{f_star_unimod_triang_last_quarter}.
A lattice polytope $P$ is \emph{Gorenstein of index $g$} if 
\begin{itemize}
\item $nP$ contains no interior lattice points for $1 \le n < g$,
\item $gP$ contains a unique interior lattice point, and 
\item $\ehr_{ P } (n-g)$ equals the number of interior lattice points in $nP$, for $n > g$.
\end{itemize}
This is equivalent to $P$ having degree $d+1-g$ and a symmetric $h^*$-vector (with respect to its degree).
\begin{theorem}\label{thm:gorenstein}
Let $P$ be a $d$-dimensional Gorenstein polytope of index $g$. Then
\[
   f^*_{k-1} >f^*_{k}
  \qquad \text{ for } \ \tfrac 1 2 \left( d + 1 + \left\lfloor \tfrac {d+1-g} 2 \right\rfloor
\right) \le k \le d \, .
\]
\end{theorem}
Going even further, for a certain class of polytopes we can prove unimodality of the
$f^*$-vector, a consequence of the following refinement of
\cref{thm:main}(\ref{f_star_unimod_triang_last_quarter}) for polytopes with degree $<
\frac d 2$.
\begin{theorem}\label{h* degree}
Let $P$ be a $d$-dimensional lattice polytope with positive degree $\le s$. Then
  \[ f^*_{k-1} > f^*_{k} \qquad \text{ for } \ \lceil\tfrac{d+s}{2}\rceil \leq k \leq
d \, , \]
unless the degree of $P$ is $0$, i.e., $P$ is a unimodular simplex with $f^*$-vector as in \cref{eg}.
\end{theorem}
This theorem implies that lattice $d$-polytopes of degree $s$ satisfying $s^2 - s - 1 \le
\frac{d}{2}$ have a
unimodal $f^*$-vector (see \cref{proposition for pyramid} below for details).
One family with asymptotically small degree, compared to the dimension, is given by taking iterated pyramids.
Given a polytope $P \subset \R^d$, we denote by $\Pyr(P) \subset \R^{ d+1 }$ the convex hull of $P$ and the $(d+1)$st unit vector.
It is well known that $P$ and $\Pyr(P)$ have the same $h^*$-vector (ignoring an extra 0), and so we conclude:
\begin{corollary}\label{pyramids}
If $P$ is any lattice polytope then $\Pyr^n(P)$ has unimodal $f^*$-vector for sufficiently
large~$n$.
\end{corollary}
% =================================
\section{{Selected proofs}}
{The aim of this section is to offer some insight into the proofs of the results stated in \cref{intro}. See \cite{fstarvectors} for full details of the proofs.

The main ingredient in the proofs of \cref{thm:main}\eqref{f_star_unimod_triang_first_half} and \eqref{f_star_0ismin} and \cref{h* degree} is the nonnegativity of $h^*$-vectors; we omit the proof details. 
\cref{pyramids} is a consequence of \cref{h* degree} together with the following proposition, whose proof again relies on  the nonnegativity of $h^*$-vectors.
}
\begin{proposition}\label{proposition for pyramid} Let $P$ be a $d$-dimensional lattice polytope that has degree at most $s$ for some $s\geq 1$.
If $d\geq2s^2-2s-2$ then the $f^*$-vector is unimodal with a (not necessarily "sharp") peak at $f^*_p$, where $\lfloor\frac{d}{2}\rfloor\leq p\leq \lceil\frac{d+s}{2}\rceil-1$.
\end{proposition}
 
The next proofs use more than just the nonnegativity of $h^*$-vectors. The first
result needs the following elementary lemma on binomial coefficients.
\begin{lemma}\label{lem:binomdiff}
Let $j, k, n$ be positive integers such that $k \le n+1-j$. Then, for $n\neq 2k-1$,
\[
  \left| \binom n k - \binom n {k-1} \right| \ \ge \ 
  \left| \binom {n-j} k - \binom {n-j}{k-1} \right| .
\]
\end{lemma}
{We are now prepared to prove \cref{thm:main}(\ref{f_star_unimod_triang_last_quarter}).}

\begin{proof}[Proof of \cref{thm:main}(\ref{f_star_unimod_triang_last_quarter})]
The inequality $f_{d-1}^*>f_d^*$ holds by \cref{h* degree}.
Now, let $ \left\lfloor \frac{3d}{4} \right\rfloor+1\leq k < d$. By (\ref{f* and h*}),
   \begin{equation}\label{part 0}f_{k-1}^*- f_{k}^* 
   \ = \ \sum_{j=0}^{k+1} \left(\binom{d+1-j}{k-j}-\binom{d+1-j}{k+1-j}\right)h_j^* \, .\end{equation}    
The difference $\binom{d+1-j}{k-j}-\binom{d+1-j}{k+1-j}$ is nonnegative whenever $k-j\geq \lfloor \frac{d+1-j}{2} \rfloor$ and negative otherwise. Hence, the difference is nonnegative whenever $j\leq 2k-d$ and negative whenever $j> 2k-d$.
Since 
$2d-2k<2k+1-d$ for $\lfloor \frac{3d}{4} \rfloor+1\leq k $,
from \eqref{part 0} we obtain 
  \begin{align}
  f_{k-1}^*- f_{k}^*
  \ &\ge \ \sum_{j=0}^{2d-2k} \left(\binom{d+1-j}{k-j}-\binom{d+1-j}{k+1-j}\right)h_j^* \label{part 1a}
 \\ &\quad + \sum_{j=2k+1-d}^{k+1} \left(\binom{d+1-j}{k-j}-\binom{d+1-j}{k+1-j}\right)h_j^*\label{part 1b}
     \end{align}
     where the differences appearing in \eqref{part 1a} are nonnegative and the ones in
\eqref{part 1b} are negative. Our aim is to compare the sums in \eqref{part 1a} and
\eqref{part 1b} to conclude that $f_{k-1}^*- f_{k}^*$ is positive.
Using standard identities 
for binomial coefficients, 
the right hand-side of \eqref{part 1a} equals
     \begin{align*}
&\sum_{j=0}^{2d-2k} \left(\sum_{l=j}^{2d-2k-1}
\left(\binom{d-l}{k-l}-\binom{d-l}{k+1-l}\right)+
\left(\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}\right) \right)h_j^* \\
 &= \sum_{l=0}^{2d-2k-1}\left( \left(\binom{d-l}{k-l}-\binom{d-l}{k+1-l}\right)
\sum_{j=0}^{2d-2k-1-l}h_j^*\right) \\
&\qquad {} + \left(\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}\right)
\sum_{j=0}^{2d-2k}h_j^* \, ,
     \end{align*}
     whence we conclude that right hand-side of \eqref{part 1a} is bounded below by 
     \begin{align}
&\left(\binom{d}{k}-\binom{d}{k+1}\right) h_0^* +
\left(\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}\right) \sum_{j=0}^{2d-2k}h_j^* \nonumber
        \\ &> \left(\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}\right) \sum_{j=0}^{2d-2k}h_j^*\label{part 5a}
    \end{align}
since $\binom{d}{k}-\binom{d}{k+1}>0$ for $\left\lfloor\frac{3d}{4} \right\rfloor+1\leq k < d$, 
and $h_0^*=1, h_j^*\geq 0$ for $j=1,...,2d-2k-1$.
On the other hand, for the differences appearing in \eqref{part 1b}, using that $2d-2k<j $ and $j\leq k+1$, it follows by \cref{lem:binomdiff}  that
\[ \left|
\binom{d+1-(2d -2k)}{d+1-k}-\binom{d+1-(2d -2k)}{d-k}
\right|\geq \left|\binom{d+1-j}{d+1-k}-\binom{d+1-j}{d-k}\right|,\]
i.e.,
\[ \left|
\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}
\right|\geq \left|\binom{d+1-j}{k-j}-\binom{d+1-j}{k+1-j}\right|.\] 
Hence for $j\geq 2k+1-d$,
\[
-\left(\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}\right)
\leq
\binom{d+1-j}{k-j}-\binom{d+1-j}{k+1-j} \, .\] 
Since both $-\binom{d+1-j}{k-j}+\binom{d+1-j}{k+1-j}$ and $h_j^*$ are nonnegative for
$j\geq 2k+1-d$, the sum in \eqref{part 1b} is bounded below by
     \begin{align}\label{last part} 
     -\left(\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}\right)
     \sum_{j=2k+1-d}^{d}h_j^* \, .\end{align}
Now \eqref{part 5a} and \eqref{last part} yield 
\[f_{k-1}^*- f_{k}^*> \left(\binom{2k-d+1}{3k-2d}-\binom{2k-d+1}{3k-2d+1}\right)
\left( \sum_{j=0}^{2d-2k}h_j^*-\sum_{j=2k+1-d}^{d}h_j^*\right).\]
Hibi \cite{hibi} showed that the inequality
\begin{equation}\label{eq:Hibi}   
\sum_{j=0}^{m+1}  h^{*}_j \ \geq \ \sum_{j=d-m}^d h^{*}_j
\end{equation} holds for $ m=0,...,\lfloor\frac d 2\rfloor-1$.
Since $ 2d-2k-1\leq\lfloor \frac{d}{2}\rfloor-1$ 
for $\left\lfloor \frac{3d}{4} \right\rfloor+1\leq k$, we can use \eqref{eq:Hibi}
to finally obtain
\[f_{k-1}^*- f_{k}^*>0 \, . \qedhere \]
\end{proof}
\begin{proof}[Proof of \cref{thm:lowdimunimodal}]
If $d=1$ or $2$, there is nothing to prove.
If $3\leq d\leq 6$, then  
by \cref{thm:main},  either
\[
f^*_{0}\leq \cdots \leq f^*_{\left\lfloor \frac{d}{2}\right\rfloor} \geq f^*_{\left\lfloor \frac{3d}{4} \right\rfloor} \geq\cdots \geq f^*_d\]
or
\[
f^*_{0}\leq \cdots \leq f^*_{\left\lfloor \frac{d}{2}\right\rfloor} \leq f^*_{\left\lfloor \frac{3d}{4} \right\rfloor} \geq\cdots \geq f^*_d.\]
{For $7\leq d\leq 13$, it suffices to check that if 
$ f^*_{i}\geq  f^*_{i+1}, $ then $f^*_{i+1}\geq f^*_{i+2}$, for all $\left\lfloor \frac{d}{2}\right\rfloor\leq i\leq \left\lfloor \frac{3d}{4} \right\rfloor-2$. By \cref{thm:main}, this will imply the unimodality of $(f_0^*,f_{1}^*,...,f_d^*)$. Using the inequality
\begin{equation}\label{eq:Hibi2}   
\sum_{j=1}^{m+1}  (h^{*}_j-h^*_{d+1-j}) \ >\ 0 \, ,
\end{equation} which holds for $ m=0,...,\lfloor\frac d 2\rfloor-1$ \cite[Remark 1.2]{stapledondelta},
 as well as the nonnegativity of $h^*$-vectors, for each value of $d$, we were able to verify that $f^*_{i+1}- f^*_{i+2}\geq c( f^*_{i}- f^*_{i+1})$ for some nonnegative real $c$ in each case.}
 

To construct a polytope with nonunimodal $f^*$-vector, we employ a
family of simplices introduced by Higashitani~\cite{higashitanicounterex}.
Concretely, denote the $j$th unit vector by $e_j$ and let
\[\Delta_w \ := \ \text{conv}\big\{0,e_1,e_2,...,e_{14},w\big\}\]
where
\[ w \ := \
(\underset{\text{7}}{\underbrace{1,1,\ldots,1}},\underset{\text{7}}{\underbrace{131,131,\ldots,131}},132)
\, .\]
It has $h^*$-vector $$
(1,\underset{\text{7}}{\underbrace{0,0,\ldots,0}},131,\underset{\text{7}}{\underbrace{0,0,\ldots,0}})$$
and, via \eqref{f* and h*}, $f^*$-vector
\begin{align*}
    & (16, 120, 560, 1820, 4368, 8008, 11440, 13001,\\
    &\quad 12488, 11676, 11704, 10990, 7896, 3788, 1064, 132) \, . \qedhere
\end{align*}
\end{proof}
{We record the following consequence of \cref{thm:lowdimunimodal}, which follows by the nonnegativity of $h^*$-vectors.}
\begin{corollary}\label{small h*} Every lattice polytope of degree at most $5$ has unimodal $f^*$-vector. 
\end{corollary}

% =================================
% \section{Gorenstein Polytopes}
\begin{proof}[Proof of \cref{thm:gorenstein}]
Let $s:=d+1-g$. We first consider the case that $s$ is odd; the case $s$ even will be
similar.
Since $h^*_j = 0$ for $j > s$ and $h^*_j = h^*_{ s-j }$, 
\begin{align*}
  &f_{ k-1 }^* - f_k^*
  \ = \ \sum_{j=0}^s \left( \binom{d-j+1}{k-j} - \binom{d-j+1}{k-j+1} \right) h_j^* \\
  &\quad= \ \sum_{j=0}^{ \lfloor \frac s 2 \rfloor } \left( \binom{d-j+1}{k-j} -
\binom{d-j+1}{k-j+1} \right) h_j^* + \sum_{j=\lfloor \frac s 2 \rfloor + 1}^s \left( \binom{d-j+1}{k-j} - \binom{d-j+1}{k-j+1} \right) h_j^* \\
  &\quad= \ \sum_{j=0}^{ \lfloor \frac s 2 \rfloor } \left( \binom{d-j+1}{k-j} -
\binom{d-j+1}{k-j+1} + \binom{d-s+j+1}{k-s+j} - \binom{d-s+j+1}{k-s+j+1} \right) h_j^* \, .
\end{align*}
Because we assume $k \ge \frac 1 2 (d + 1 + \lfloor \frac s 2 \rfloor)$,
\[
  \binom{d-j+1}{k-j} - \binom{d-j+1}{k-j+1} \ > \ 0 
\]
for $0 \le j \le \lfloor \frac s 2 \rfloor$.  
The inequality  \[ \binom{d-j+1}{k-j} - \binom{d-j+1}{k-j+1} + \binom{d-s+j+1}{k-s+j} -
\binom{d-s+j+1}{k-s+j+1}>0\] follows directly if
 $\binom{d-s+j+1}{k-s+j} -
\binom{d-s+j+1}{k-s+j+1}\geq 0$ or $k-s+j+1<0$.
Otherwise, \cref{lem:binomdiff} implies that, for the same range of $j$,
\[
  \binom{d-j+1}{k-j} - \binom{d-j+1}{k-j+1} + \binom{d-s+j+1}{k-s+j} -
\binom{d-s+j+1}{k-s+j+1} \ \ge \ 0 \, .
\]
In fact, the last inequality is strict for $k \ge \frac 1 2 (d + 1 + \lfloor \frac s 2 \rfloor)$ {(the proof is analogous to that of \cref{lem:binomdiff}).}
Finally we use that $h_j^* \ge 0$ and $h_0^*=1$ to deduce that $f_{ k-1 }^* - f_k^* > 0$.
The computations in the case $s$ even is very similar. Now we write
\begin{align*}
  &f_{ k-1 }^* - f_k^*
  \ = \ \sum_{j=0}^s \left( \binom{d-j+1}{k-j} - \binom{d-j+1}{k-j+1} \right) h_j^* \\
  &\quad= \ \sum_{j=0}^{ \frac s 2 - 1 } \left( \binom{d-j+1}{k-j} - \binom{d-j+1}{k-j+1} +
\binom{d-s+j+1}{k-s+j} - \binom{d-s+j+1}{k-s+j+1} \right) h_j^* \\
  &\quad \qquad {} + \left( \binom{ d - \frac s 2 + 1 }{ k - \frac s 2 } - \binom{ d -
\frac s 2 + 1 }{ k - \frac s 2 + 1 } \right) h_{ \frac s 2 }^*
\end{align*}
and use the same argumentation as in the case $s$ odd.
\end{proof}
% =================================
\section{Concluding Remarks}
There are many avenues to explore $f^*$-vectors, e.g., along analogous studies of
$h^*$-vectors, and we hope the above results form an enticing starting point.
% We conclude with a few open questions which are apparent from the above.
The techniques in our proof of \cref{thm:lowdimunimodal} do not offer much insight
in the case of $14$-dimensional lattice polytopes as there are candidates $f^*$-vectors
with corresponding  $h^*$-vectors that satisfy all inequalities discussed in
\cite{stapledondelta}. It is unknown, though, if such polytopes exist.

Higashitani~\cite[Theorem 1.1]{higashitanicounterex} provided examples of $d$-dimensional
polytopes with nonunimodal $h^*$-vector for all $d\geq3$. Therefore, by
\cref{thm:lowdimunimodal} we have examples of polytopes that have such an
$h^*$-vector but their $f^*$-vector is unimodal. It would be interesting to know if the
opposite can be true, that is, if there exist polytopes with unimodal $h^*$-vector and
nonunimodal $f^*$-vector. By \cref{small h*}, such polytopes would need to have degree at least~6.

Whenever one detects that a given given is unimodal, it is natural to ask about the
stronger property that the polynomial is log concave or, even stronger, real rooted. Our
methods do not yield these properties but it would be interesting if one could extend,
e.g., \cref{pyramids} or \cref{proposition for pyramid} along these
lines.

Finally, starting with Stapledon's work~\cite{stapledondelta}, there has been much recent
attention to symmetric decompositions of $h$- and $h^*$-polynomials;
see, e.g.,~\cite{athanasiadisinterior,hstarboundary} and, in particular,~\cite{brandensolus}
where analogous decompositions for $f$-vectors are discussed. We believe this line of
research is worthy of attention with regards to understanding $f^*$-vectors and the inequalities that hold among their coefficients.

\acknowledgements{We thank the organizers of \emph{Research Encounters in Algebraic and Combinatorial
Topics} (REACT 2021), where our collaboration got initiated. We are grateful to Michael
Joswig, Matthias Schymura, Liam Solus and Lorenzo Venturello for helpful conversations, and to Eleni Tzanaki for useful comments on the Greek abstract, and the anonymous referees for the careful reading and suggestions.}

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