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Find an example of a bounded convex set $S$ in $\mathbb{R}^{2}$ such that its profile $P$ is nonempty but conv $P \neq S$

Take any square as $S$ such that top and left sides are included and bottomand right sides are not included in $S$ .Then, $P$ contains three points. Bottom left vertex, top left and top rightvertex, and convex hull of $P$ is triangle with those points as vertices.(not that here I use term vertex as it is defined in geometry, not here inlinear algebra)

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 5

Polytopes

Vectors

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

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were asked to find an example of a bounded conduct set s in our to such that the profile of set is non empty. But the convex hole of the profile is not the same as the set. So this is different from a previous problem in the previous problem. Exercise 10. We're looking for an example of a compact contact set with this property, but now we're looking for a bounded convex set. So an easy example is to consider a square. We're only two of the sides are included in the set, So we're going to let s be a square with just two adjacent edges. In other words, it's going to look something like this. The dotted lines indicate that this boundary is not included. Now we see that the context hole Well, Katie profile is non empty because mm profile P includes the three. Yeah. Mm hmm. Corners shall guess I'll put in quotes because technically, two of them aren't really corners. But the three corners of this square, So the profile looks something like this is the only extreme points. So you have these three points for the profile, and we have that the convex hole of P is a triangle, so forming the convex hole which are drawing green, you simply draw a triangle between these three points and it's clear at the convicts. Whole of P is not the same as the set s.

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