Shadow systems, decomposability and isotropic constants

TL;DR

This work analyzes local maximizers of the isotropic constant $L_K$ in high dimensions through Minkowski and RS-type decomposability. By developing a second-derivative framework for isotropic variations and introducing shadow systems and generalized RS-movements, it proves that the polar body $K^\circ$ of any local maximizer has bounded decomposability, namely $\dim \mathcal{S}(K^\circ) \le \frac{n^2+3n}{2}$, linking extremality to rigidity. The paper also explores the polytopal case via facewise affine structures and connects to affine rigidity, and it shows symmetry-based refinements that tighten the bound in several natural classes. These results illuminate how rigidity, symmetry, and combinatorial structure constrain extremizers of $L_K$ and open avenues for sharpening the bound or studying Minkowski decomposability directly. Overall, the work clarifies structural constraints on local maximizers of the isotropic constant and lays groundwork for deeper decomposability analyses in convex geometry.

Abstract

We study necessary conditions for local maximizers of the isotropic constant that are related to notions of decomposability. Our main result asserts that the polar body of a local maximizer of the isotropic constant can only have few Minkowski summands; more precisely, its dimension of decomposability is at most $\frac12(n^2+3n)$. Using a similar proof strategy, a result by Campi, Colesanti and Gronchi concerning RS-decomposability is extended to a larger class of shadow systems. We discuss the polytopal case, which turns out to have connections to (affine) rigidity theory, and investigate how the bound on the maximal number of irredundant summands can be improved if we restrict our attention to convex bodies with certain symmetries.

Figures (2)

• Figure 1: An illustration of Definition \ref{['def_generalized_RS_movement']} in the case where $K$ is a polygon. In Subfigure (A), the red area corresponds to the set where $\beta^+$ and $\beta^-$ have to be equal by condition \ref{['def_generalized_RS_movement_2']}. In Subfigure (B), the sets $K_{0.1} \setminus K$ and $K_{0.2} \setminus K$ are shaded in cyan, whereas the sets $K \setminus K_{0.1}$ and $K \setminus K_{0.2}$ are shaded in orange.
• Figure 2: An illustration of Construction \ref{['constr_K_t']} in the case where $K$ is a polygon. In Subfigure (A) and Subfigure (C), the sets $K_{-\varepsilon} \setminus K$ and $K_{\varepsilon} \setminus K$ are shaded in cyan, whereas the sets $K \setminus K_{-\varepsilon}$ and $K \setminus K_{\varepsilon}$ are shaded in orange. In this example, as in the polytopal case more generally, the vertices move on rays emanating from the origin (marked in blue); their distance to the origin at time $t$ is the weighted harmonic mean of the initial and the final distance.

Theorems & Definitions (42)

• Theorem 1.1: Meyer-Reisner
• Theorem 1.2
• Definition 1.3
• Theorem 1.4: Campi-Colesanti-Gronchi
• Theorem 1.5
• Definition 2.1
• Proposition 2.2: Rademacher
• proof : Proof of Proposition \ref{['prop_rademacher']}
• Corollary 2.3
• Definition 2.4