Prove that when a sequence of convex bodies converges to a convex body in the Hausdorff metric, the sequence can be bounded within scaled versions of the limit body | Step-by-Step Solution

Hello! This is a beautiful problem that connects metric topology with geometric analysis.

What We're Solving:

We need to prove that if a sequence of convex bodies {Kₙ} converges to a convex body K in the Hausdorff metric, then there exist positive constants c and C such that for sufficiently large n: cK ⊆ Kₙ ⊆ CK

This is essentially showing that convergent sequences of convex bodies are "uniformly bounded" in terms of scaling.

The Approach:

This is like a "sandwich theorem" for convex bodies! We're showing that all the bodies in our sequence eventually get trapped between a smaller and larger scaled version of the limit body. The key insight is that Hausdorff convergence gives us control over how far the bodies can deviate from each other, and convexity helps us translate this into scaling bounds.

Step-by-Step Solution:

Step 1: Recall what Hausdorff convergence means Since Kₙ → K in Hausdorff metric, for any ε > 0, there exists N such that for n ≥ N:

• Kₙ ⊆ K + εB (every point in Kₙ is within ε of some point in K)
• K ⊆ Kₙ + εB (every point in K is within ε of some point in Kₙ)

where B is the unit ball.

Step 2: Use convexity to establish inner bounds Since K is convex and contains the origin in its interior (we can always translate), there exists r > 0 such that rB ⊆ K.

From K ⊆ Kₙ + εB, we get that every point in rB is within ε of some point in Kₙ. For small enough ε (specifically, ε < r), this forces (r-ε)B ⊆ Kₙ.

Step 3: Scale to get the inner containment Since (r-ε)B ⊆ Kₙ and rB ⊆ K, we have: ((r-ε)/r)K ⊇ ((r-ε)/r)rB = (r-ε)B ⊆ Kₙ

So we get cK ⊆ Kₙ with c = (r-ε)/r.

Step 4: Establish outer bounds using boundedness Since K is bounded, it's contained in some ball RB. From Kₙ ⊆ K + εB ⊆ RB + εB = (R+ε)B.

Step 5: Scale to get outer containment Since K contains rB and Kₙ ⊆ (R+ε)B, we have: Kₙ ⊆ (R+ε)B = ((R+ε)/r) · rB ⊆ ((R+ε)/r)K

So we get Kₙ ⊆ CK with C = (R+ε)/r.

The Answer:

For any sequence of convex bodies {Kₙ} converging to convex body K in Hausdorff metric, there exist constants 0 < c < 1 < C such that for sufficiently large n: cK ⊆ Kₙ ⊆ CK

The constants can be taken as c = (r-ε)/r and C = (R+ε)/r, where r is the inradius of K, R is its circumradius, and ε is sufficiently small.

Memory Tip:

Think "Hausdorff sandwich"! Hausdorff convergence gives you ε-neighborhoods, convexity lets you convert these to radial bounds, and scaling creates the sandwich. The convex bodies get "squeezed" between scaled versions of the limit body! 🥪

This result is fundamental in convex geometry because it shows that Hausdorff convergence is "well-behaved" for convex bodies - they can't suddenly become arbitrarily large or small.