Proof of a conjecture of Ehrenborg on the pizza quantity

摘要: Define the polynomials $F_m(x,y)$ by the formal generating function
\[
 \sum_{m\ge0}F_m(x,y)\frac{t^m}{m!}
 =\frac{1-(x+y)t}
 {\sqrt{1-2(x+y)t}\,
  \bigl(1-2(x+y)t+(x-y)^2t^2\bigr)}.
\]
We show that $F_m(x,y)$ is $\gamma$-positive, which proves Ehrenborg's conjecture on the homogeneous terms in the Taylor expansion of the pizza quantity for a three-dimensional ball cut by two perpendicular planes, the alternating sum of the volumes of the four resulting regions. A convolution identity involving Chebyshev polynomials of the first kind is also established.  For fixed $x,y\ge0$ with $x+y>0$, we prove that the sequence $\{F_m(x,y)\}_{m\ge0}$ is strictly log-convex.