About convex hull and fixed point

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SUMMARY

The discussion focuses on the mathematical concepts of convex hulls and fixed points within the context of the set X=[0,1]^2. The user explores the set a(x) defined as a(x)={y in X:||y-x||>=1/4} and seeks to identify the fixed points, concluding that they lie within the bounds 3/4>=x>=1/4 and 3/4>=y>=1/4. Additionally, the user examines a(x)={y in X:||y-x||>=1/2}, determining that the fixed points are located at x=1/2 or y=1/2. The discussion also includes inquiries about the definitions of convex hull and fixed point.

PREREQUISITES
  • Understanding of convex hulls in geometry
  • Knowledge of fixed point theory
  • Familiarity with Euclidean distance metrics
  • Basic concepts of set theory
NEXT STEPS
  • Study the properties of convex hulls in two-dimensional spaces
  • Research fixed point theorems, particularly in metric spaces
  • Explore applications of convex analysis in optimization problems
  • Learn about distance functions and their implications in geometry
USEFUL FOR

Mathematicians, students of geometry, and anyone interested in advanced mathematical concepts related to convex analysis and fixed point theory.

sapporozoe
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First, I wonder whether I can put the post here...

Given
X=[0,1]^2

a(x)={y in X:||y-x||>=1/4}

b(x)is the convex hull of a(x).

Identify the set of fixed points.

My answer is 3/4>=x>=1/4, 3/4>=y>=1/4, but I am not sure...

What if we have a(x)={y in X:||y-x||>=1/2}? (My answer is x=1/2 or y=1/2)

Thanks.
 
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What is the definition of convex hull? Of fixed point?
 

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