Rounding
For a center \(c\) and an origin-centered ellipse \(E\), the ellipse \(c+E\) is a \(\rho\)-rounding of \(K\) if \(E\subseteq K-c\subseteq \rho E\). Each panel keeps the center and the shape computed by its problem and shows the smallest such \(\rho\). Writing \(E=\{Lu:\|u\|\le 1\}\) for a polygon with vertices \(v\) and facets \(a_i^\top x\le b_i\),
\[\rho=\frac{r_{\rm out}}{r_{\rm in}},\qquad r_{\rm out}=\max_{v}\big\|L^{-1}(v-c)\big\|,\qquad r_{\rm in}=\min_i \frac{b_i-a_i^\top c}{\|L a_i\|}.\]
The two drawn ellipses are \(c+r_{\rm in}E\) and \(c+r_{\rm out}E\). The rounded view applies \(x\mapsto L^{-1}(x-c)/r_{\rm in}\), which turns them into circles of radius \(1\) and \(\rho\). Figure 1 of the paper draws the companion ellipses of (c) and (d) at the scalings proved in Lemmas 2.1 and 4.1 (factors 1.6679 and 1.6720); this page draws the tightest scaling (1.6317 and 1.6574).
Symmetry
\(\mathrm{sym}(x,K)=\max\{\alpha\ge 0: -\alpha(K-x)\subseteq K-x\}\) and \(\mathrm{sym}(K)=\max_{x}\mathrm{sym}(x,K)\in[1/n,\,1]\); a maximizer is a Minkowski center. The value is \(1/n\) exactly for simplices and \(1\) exactly for centrally symmetric bodies. For a polygon, one linear program gives both (Belloni and Freund, 2008): maximize \(\beta\) subject to \(a_i^\top c+\beta\,(b_i-\min_{y\in K}a_i^\top y)\le b_i\) for all \(i\); then \(\mathrm{sym}(K)=\beta/(1-\beta)\).
The four problems
Move a Minkowski center to the origin, set \(\alpha=\mathrm{sym}(K)\) and \(\lambda=\frac{1+\alpha}{1-\alpha}\), and write \(K=\mathrm{conv}\{v_j\}=\{x: h_i^\top x\le 1\}\). Then (a) and (b) are the classical problems, and (c) and (d) add terms that involve \(\lambda\):
\[\begin{aligned}
&\text{(a) MVCE:} && \min_{Q\succ 0,\,c}\; -\log\det Q && \text{s.t. } (v_j-c)^\top Q (v_j-c)\le 1,\\
&\text{(b) MVIE:} && \min_{X\succ 0,\,c}\; -2\log\det X && \text{s.t. } \|Xh_i\|\le 1-h_i^\top c,\\
&\text{(c) } (\mathrm P_\lambda): && \min_{Q\succ 0,\,c}\; -\log\det Q && \text{s.t. } (v_j-c)^\top Q (v_j-c)+\lambda\, c^\top Q c\le 1,\\
&\text{(d) } (\mathrm J_\lambda): && \min_{X\succ 0,\,c}\; -2\log\det X && \text{s.t. } \|Xh_i\|^2+(\lambda-1)(h_i^\top c)^2+2h_i^\top c\le 1.
\end{aligned}\]
The ellipse is \(c+\{x: x^\top Qx\le 1\}\) in (a), (c) and \(c+X B\) in (b), (d). The extra terms pull the center toward the Minkowski center, more strongly when \(K\) is more symmetric. At \(\alpha=1\), \(\lambda=\infty\) fixes \(c=0\).
Guarantees
(a) and (b) give \(\rho\le n\) for every convex body and \(\rho\le\sqrt n\) for centrally symmetric ones (John, 1948), and these factors are sharp. (c) and (d) give \(\rho\le\sqrt{n/\mathrm{sym}(K)}\) (Theorem 1.1, Lemmas 2.1 and 4.1), as conjectured by Belloni and Freund in 2005. This bound equals \(n\) for a simplex and \(\sqrt n\) for a centrally symmetric body, and Theorem 5.1 shows that it cannot be improved by more than the factor \(\sqrt{2/3}\) for any \(n\) and any symmetry value. All of these quantities are unchanged by linear changes of coordinates. The page draws the example of Figure 1 after such a change, so the picture differs from the printed one while every number agrees.
The slider family
From \(\mathrm{sym}=1/2\) to \(0.7072\), the slider cuts two corners off a triangle (dotted in panel (b)) and ends at the polygon of Figure 1. Every polygon on this part lies inside the triangle and contains the triangle's maximum-volume inscribed ellipse and one of its corners, so the MVIE of the polygon is the triangle's inscribed ellipse and (b) stays at exactly \(\rho=2\). From \(0.7072\) to \(1\), a smooth path leads to the unit square. We chose this path numerically: among pentagons that contain three corners of the square, it keeps (c) and (d) as far ahead of (a) and (b) as we could while \(\mathrm{sym}(K)\) increases. Other polygons behave differently; for some, a classical ellipse is the roundest (try Draw your own).
Computation
Everything is computed live in the browser. The symmetry LP has three variables, so the page enumerates its vertices. The four log-det problems are solved by one primal barrier method with damped Newton steps in five variables (the shape and the center), in about 1 ms per polygon. The solver was checked against MOSEK on 67 polygons (largest relative difference in \(\rho\) 1.6e-4, within MOSEK's tolerance) and against the exact values for the triangle, the square, and the regular hexagon (error below 1e-15).