1. Ehrhart Positivity

Conjecture 1.05.
Let $[n]:=\{1,\ldots,n\}$ be a groundset, and let $i(P; t)$ denote the Ehrhart polynomial of a lattice polytope $P$. Then coefficientwise, $$i(T_{r,n}; t) \leq i(\mathsf{Pan}_{s,r,n};t) \leq i(U_{r,n};t),$$ where $T_{r,n}$ is the rank $r$ minimal matroid on $[n]$, $U_{r,n}$ is the rank $r$ uniform matroid on $[n]$, and $\mathsf{Pan}_{s,r,n}$ is the rank $r$ Panhandle matroid on $[n]$. 
Problem 1.1.
What is known (or can be shown) about the Ehrhart positivity of polytopes related to Birkhoff polytopes (e.g., transportation polytopes)? 
Problem 1.15.
Are $d$dimensional smooth polytopes with $\leq d+3$ facets Ehrhart positive or satisfy the higher integrality condition? 
Problem 1.2.
 (a) Is there a closed formula for the Ehrhart polynomial of order polytopes of fence posets? Further, are these Ehrhart polynomials Ehrhart positive?
 (b) Is the $h^\ast$polynomial of the order polytopes from (a) realrooted?
 (c) Can this be extended to generalized permutahedra?

Problem 1.25.
Does a rational polygon $P\subseteq \R^2$ with quasiperiod 1 satisfy Scottâ€™s inequality? That is, assuming $i\geq 1$, is the inequality $b\leq 2i+7$ satisfied, where $b$ and $i$ represent the number of boundary and interior lattice points of $P$, respectively. 
Problem 1.3.
Let $ASM_n$ denote the $n$th alternating sign matrix polytope (defined as the convex hull in $\mathbb{R}^{n^2}$ of the $n\times n$ alternating sign matrices). It is known that $i(ASM_n;t)$ is Ehrhart positive. What further can be said about $i(ASM_n;t)$? Furthermore, what can be said about $h^*(ASM_n;z)$? 
Definition: Given two lattice polytopes $P\subseteq \R^d$ and $P'\subseteq\R^{d'}$, the join of $P$ and $P'$ is given by $$P*P' := \mathrm{conv}\{(P,0_{d'},1) \cup (0_{d},P',0)\}.$$
Problem 1.35.
For lattice polytopes $P$ and $P'$, which Ehrharttheoretic properties pass through their join? For example, if $P$ and $P'$ are both Ehrhart positive, is $P*P'$ as well? 
Problem 1.4.
In types B, C, and D, the stretched ClebschGordon coefficients are period2 quasipolynomials. Are the coefficients of the constituent polynomials always positive? 
Problem 1.45.
Consider rational polygons of denominator 2. The Ehrhart quasipolynomial has two constituent polynomials, one of which is an Ehrhart polynomial. Classify the other constituent polynomial.
Note that this is settled for up to 2 interior lattice points. 
Problem 1.6.
 (a) Consider the Ehrhart polynomial of a lattice polytope $P$. Below are two wellknown representations of $i(P;t)$:
$$i(P;t) = \sum_{i=1}^d c_it^i = \sum_{i=0}^d h_i^* \binom{t+di}{d}.$$
Fix $k\in \Z_{>0}$ and write $i(P;t)$ in the basis $\{(tk)^i\}_i$. (For example, if $\deg(h^*(P;z)) = s$, then $i(P;t)$ is nonnegative in the basis $\{(ts+1)^i\}_{i=0}^{d}$.)
Investigate what happens when changing $k$ from 0 to 1 for nonEhrhart positive polytopes (or other $k$).  (b) If $P$ is Ehrhart positive in the standard basis, is it Ehrhart positive in the basis $\{t^i(t+1)^{di}\}$?
 (a) Consider the Ehrhart polynomial of a lattice polytope $P$. Below are two wellknown representations of $i(P;t)$:
$$i(P;t) = \sum_{i=1}^d c_it^i = \sum_{i=0}^d h_i^* \binom{t+di}{d}.$$
Fix $k\in \Z_{>0}$ and write $i(P;t)$ in the basis $\{(tk)^i\}_i$. (For example, if $\deg(h^*(P;z)) = s$, then $i(P;t)$ is nonnegative in the basis $\{(ts+1)^i\}_{i=0}^{d}$.)
Cite this as: AimPL: Ehrhart polynomials: inequalities and extremal constructions, available at http://aimpl.org/ehrhartineq.