Extremal functions for real convex bodies: simplices, strips, and ellipses

Abstract

We present an explicit method to compute the (Siciak-Zaharjuta) extremal function of a real convex polytope in terms of supporting simplices and strips. We use this to give a new proof of the existence of extremal ellipses associated to the extremal function of a real convex body.

Figures (4)

• Figure 1: A solid line indicates a direct link, while a dashed line uses previously established links.
• Figure 2: Theorem \ref{['thm:simplices']} in ${\mathbb R}^3$, with $K=S_1\cap S_2$ and $S_1=\hbox{co}\{p_0,p_1,p_2,p_3\}$.
• Figure 3: Examples of strips in ${\mathbb R}^3$.
• Figure 4: Theorem \ref{['thm:main']} covers quadrilateral BCSE in the left picture (a base case). In the middle picture, let $K$ be the pentagon FGHIJ, and let $K_0$, $K_1$ be the quadrilaterals KGHI, FGLJ respectively, so that $K_2$ is the triangle KGL. Then $V_K=\max\{V_{K_0},V_{K_1}\}$. In the right picture, the intersection of the triangles $T_1=\text{MNO}$ and $T_2=\text{PQR}$ is convex but their union is not. We cannot conclude that $V_{T_1\cap T_2}=\max\{V_{T_1},V_{T_2}\}$ . (By inscribing ellipses in $T_1\cap T_2$, it is easy to see that this formula is false).

Theorems & Definitions (117)

• Theorem : Theorems \ref{['thm:main']} & \ref{['thm:mainN']} (special cases) and Theorem \ref{['thm:maingen']}
• Theorem : Theorem \ref{['prop:19']}
• Theorem : Theorem \ref{['thm:blm']}
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