Tropical Ehrhart theory and tropical volume

Abstract

We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions.

Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling

“…A classically convex tropical polytope is called a polytrope. Definition 2.7 (Covector Decomposition (From [11] and [29])). A tropical polytope may be decomposed into a polyhedral complex of polytropes known as a covector decomposition of P , denoted as C P .…”

“…The union of (e − 1)-dimensional polytropes is described in the following definition. Definition 2.16 (i-trunk and i-tentacles (Definition 2.1 in [29])). Let P be a tropical polytope and let i ∈ [e − 1] where [e − 1] = {1, .…”

“…Example 3.10 (Borrowed from [29]). Consider a tropical polytope P := tconv ({(0, −2, 5), (0, −2, 3), (0, 2, 2), (0, 1, 0)}).…”

“…Proof. The T r e−1 (P ), of P represents a tropical polytope of dimension (e − 1) [29]. Any tropical polytope, P , where dim(P ) = (e − 1) contains a B R (P ).…”

The tropicalization of the entropic barrier

“…We remark that the quantity n i=1 x * i (µ) which appears in Lemmas 5 and 6 corresponds to the tropical barycentric volume, introduced by Loho and Schymura in [LS20], of the tropical polytope {x ∈ val P : val c, x T α}. We refer to [LS20] for further properties on the tropical barycentric volume and its relation with the Euclidean volume.…”