Tropical Carathéodory with Matroids

NEW

Problem: The natural formalization is the signed-tropical analogue of Loho--Sanyal Theorem 1.3:

Let $M=(E,\mathcal I)$ be a matroid, $V:E\to \mathbb T_{\pm}^d$, and let $\operatorname{tconv}_{\pm}$ be the signed tropical convex hull of Loho--Végh. If

for every basis $B$ and every cocircuit $C$ of $M$, must there exist a basis $B_0$ with

Here $\mathbb O=(-\infty,\ldots,-\infty)$. This is the literal replacement of ordinary tropical convexity in Theorem 1.3 by signed tropical convexity. Since the question asks whether Theorems 1.3 and 2.11 can be generalized, falsifying this analogue already gives a negative answer to the combined question.

Result: No. The signed analogue of Theorem 1.3 is false.

Let $M=U_{2,5}$, the uniform matroid of rank $2$ on

Its bases are the $2$-subsets and its cocircuits are the $4$-subsets.

Work in $\mathbb T_{\pm}^2$, and set

For signed tropical convexity, $\mathbb O\in\operatorname{tconv}_{\pm}(S)$ iff there are normalized coefficients $\lambda_s\in\mathbb T_{\max}$, not all $-\infty$, such that the signed tropical sum in each coordinate is balanced.

The following triples already contain $\mathbb O$ in their signed tropical convex hulls:

For example, for $(a,b,c)$ with coefficients $(0,-2,-1)$, the two coordinate sums are balanced:

The other three rows are checked identically.

Every $4$-subset of $E$ contains one of these four triples:

Hence every cocircuit $C$, and therefore every $B\cup C$, satisfies

But no basis works. Indeed, for two nonzero signed tropical points $p,q\in\mathbb T_{\pm}^2$, the origin lies in their signed tropical convex hull only if the signs are opposite in both coordinates and the required coefficient difference is the same in both coordinates. Checking the ten pairs:

• most pairs fail because some coordinate has the same sign;
• the only pairs with opposite signs in both coordinates are $(a,b)$, $(a,c)$, and $(d,e)$;
• their valuation differences are respectively $(2,1)$, $(0,1)$, and $(1,0)$, so none can be balanced simultaneously in both coordinates.

Thus

for every basis $B$ of $U_{2,5}$.

So the hypothesis of the signed analogue of Theorem 1.3 holds, but the conclusion fails.

Citation: Definitions of signed tropical convexity are from Loho--Végh, Signed tropical convexity, arXiv:1906.06686. The theorem being generalized is Loho--Sanyal, Tropical Carathéodory with Matroids, arXiv:1912.11262. The counterexample above is self-contained.