Shifted symmetric $\delta$-vectors of convex polytopes

A $\delta$-vector $\delta (\text{Pc})=({\delta }_0,{\delta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\delta }_d)$ is called shifted symmetric if ${\delta }_{d-i}={\delta }_{i+1}$ for each $0 \leq i \leq [(d-1)/2]$\mathrm{.}Anaturalfamilyof$(0,1)$-polytopeswithshiftedsymmetric$\delta$-vectors will be studied.