Shifted symmetric -vectors of convex polytopes

Let P ⊂ R N be an integral convex polytope of dimension d and δ ( P ) = ( δ 0 , δ 1 , … , δ d ) its δ -vector. It is known that ∑ j = 0 i δ d − j ≤ ∑ j = 0 i δ j + 1 for each 0 ≤ i ≤ [ ( d − 1 ) / 2 ] . A δ -vector δ ( P ) = ( δ 0 , δ 1 , … , δ d ) is called shifted symmetric if ∑ j = 0 i δ d − j = ∑ j = 0 i δ j + 1 for each 0 ≤ i ≤ [ ( d − 1 ) / 2 ] , i.e., δ d − i = δ i + 1 for each 0 ≤ i ≤ [ ( d − 1 ) / 2 ] . In this paper, some properties of integral convex polytopes with shifted symmetric δ -vectors will be studied. Moreover, as a natural family of those, ( 0 , 1 ) -polytopes will be introduced. In addition, shifted symmetric δ -vectors with ( 0 , 1 ) -vectors are classified when ∑ i = 0 d δ i ≤ 5 .