diff --git a/metric_entropy_duality_counterexamples_v2.txt b/metric_entropy_duality_counterexamples_v2.txt
index 7ccaae2..12d3322 100644
--- a/metric_entropy_duality_counterexamples_v2.txt
+++ b/metric_entropy_duality_counterexamples_v2.txt
@@ -6,6 +6,12 @@ body K the polar is K^o = { y : <x,y> <= 1 for every x in K }, again an
 origin-symmetric convex body. All logarithms are natural; the base affects
 only constants and is immaterial.
 
+Concrete pictures ground the definitions. Covering a disk of radius R by
+translations of one square of side two takes on the order of R^2 copies, so N
+grows like the square of the radius. For the cube L = [-1,1]^n the polar is
+L^o = { y in R^n : sum_i |y_i| <= 1 }, the cross-polytope and the unit ball of
+the l1 norm, so polarity exchanges the sup norm and the l1 norm.
+
 Passing to the polar exchanges a body with the collection of linear
 functionals bounded by one on it. The duality conjecture for covering numbers
 asks whether this exchange preserves covering complexity up to a universal
@@ -36,10 +42,11 @@ Because the polar inequality can be forced to fail with L a cube, no universal
 (a, b) survives.
 
 The surrounding landscape fixes what is and is not already known. Konig and
-Milman proved that the n-th roots of the two covering numbers are comparable, which on the logarithmic scale leaves an additive error of
-order n. Bourgain, Pajor, Szarek and
-Tomczak-Jaegermann obtained entropy-number comparisons under geometric or
-sequence-regularity hypotheses, and estimates whose loss depends on the rank.
+Milman proved that the n-th roots of the two covering numbers are comparable,
+which on the logarithmic scale leaves an additive error of order n. Bourgain,
+Pajor, Szarek and Tomczak-Jaegermann obtained entropy-number comparisons under
+geometric or sequence-regularity hypotheses, and estimates whose loss depends
+on the rank.
 Thus a single universal pair for all bodies and dimensions was open.
 
 Available at ./bourgain-pajor-szarek-tomczak-1989-duality-entropy-numbers-ihes-m89-01.pdf