Convex Geometry - Science topic

A search with keywords "weak fixed point property" (which is the official name of the property you are interested in) and with "weak normal structure" (which is a widely used sufficient condition for this property) may give you a lot of information on the subject.

In addition to the very helpful answer given by @ Vladimir Kadets, there is a bit more to add.

A good paper directly related to the topic for this thread is

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Recall that a convex polytope is the convex hull of a finite set of points (for more about this, see page 2). For the Minkowski sum of polytopes, see page 20.

This is a very interesting question.

The Minkowski sum of convex hulls is one of the highlights of the following paper:

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Recall that a convex polytope is the convex hull of a finite set of points. For more about this, see page 4. For the details concerning the Minkowski sum of polytopes, see page 20.

Dear Herve, I'am interested on your topic: do you have a paper/document on theorem's proof? Thanks.  Gianluca

This is a great question.

Here is an article that may be of interest to followers of this thread:

Consider, for example, the convex hull for Steiner's Roman surface shown in Fig. 2 on page 6.

You may see: 

[1] Bryant, V. W. Independent axioms for convexity. J. of Geometry, 5, 1974, 95-99

[2]n Bryant, V.W. and Webster R.J. Convexity spaces I: J. Math.Anal.Appl. 37, 1972, 206-215, II (the same journal) 1973, 321-327; III 57 1977, 382-392

To use an analogy. Look at the relationship between discrete and continuous Hopfield model (Naykin S. Neural networks ....).

Dear Enas,

A partially answer (working in a particular case) might be the following one. Assume that you have a partially ordered ring A, which is also a vector lattice over the real field (example: A=C(X), the space of all real continuous functions on a topological space X. The positive cone P is the cone of all nonnegative functions at each point of X). Define a positive convex ideal as being a convex ideal for which all its elements are positive. Then the sum of two positive convex ideals is a positive convex ideal. In fact, if 0<=a<=x+y, x being in the first positive convex ideal and y in the second one, then using Riesz property (which works in any vector lattice), one deduces the existence of b,c, such that: a=b+c, 0<=b<=x, 0<=c<=y. Application of the definition of a convex ideal leads to the desired conclusion.   

A point is a vertex if and only if it is the basic solution for a feasible tableau. For a given point, you can check it in polynomial time (Peter T Breuer  method works in non-degenerate case). But the number of vertices could be exponential. Then in linear time you can find all adjoint vertices. If they are on your list, you are done.  The boiundness is not needed.

You can refer to the followin papers=

1) Upper-Lipschitz multifunctions and inverse subdifferentials

Roxin Zhang, Jay Treiman,

Nonlinear Analysis 01/1995; 24(2):273-286. · 1.61 Impact Factor

2)

Characterization of metric regularity of subdifferentials

Francisco Javier Aragón Artacho, Michel H. Geoffroy

Journal of Convex Analysis (Impact Factor: 0.59). 01/2008; 15(2):365-380.

Dear James Peters,

Yes, there are many applications of convexity, or convex sets, which sounds very similar to Christopher Alexander's positive space, one of the 15 structural properties he developed in The Nature of Order: http://en.wikipedia.org/wiki/The_Nature_of_Order

People tend to perceive things from this perspective of convexity in order to see identifiable pieces of a whole. This perspective (focusing on individuals) however is somehow contradictory to the holistic (or fractal) perspective. For example, drop a wine glass on the ground, and it is very likely broken into many pieces. These many pieces are actually mixed up with convex and concave ones. This mixed up is the same for natural cities we identified or detected using massive geographic information; see related paper: http://arxiv.org/ftp/arxiv/papers/1501/1501.03046.pdf

Having said so, may I draw a tentative conclusion as such? The perspective of convexity is Euclidean, focusing on individual, relying on human perception, whereas the fractal or holistic perspective is not necessary to be convex only, but mixed up with convex and concave ones. For example, the math snowflake is concave in essence, and is made of many convex triangles of course. 

This is one of those questions with a somewhat narrow perspective.    Still, however, the underlying problem of approximating a surface with a Delaunay triangulation or a Dirichlet tessellation (tessellating a region of space with Voronoi regions of points is named after Dirichlet) is important and well-worth considering.

A good starting point in looking for an answer to your question is to consider

H.S. Muddana, Integrated biomechanical model of cells embedded in extracellular matrix, M.S. thesis, Texas A&M University, 2006:

See, for example, the introduction to Delaunay triangulation and its generalization, starting on page 26 (bottom of the page).    See the discussion about mesh refinement, starting on page 48.

Very nice example!

I'll try to pay you back by reading your papers with Cascales and Rodríguez, which have been on my table for quite some time.

It is always lovely when you can form a convex polyhedron with a discrete optimization problem.  It gives you properties you can exploit, some that you have pointed out.  The application I'd like to point out is Scheduling.  Scheduling is pretty notorious for exploiting properties of convex polyhedra if the problem as an optimization problem can be formulated as a relaxed LP (of its IP counterpart).  Of course this usually means solving a linear program then rounding can be used (or in some realms is avoided, but instead primal-dual approaches are found at times).  

Here is an example of how (the rounding theorem in this paper is the crux of how):  https://www.researchgate.net/publication/4355176_Approximation_algorithms_for_scheduling_unrelated_parallel_machines

The above link is to a very important 2-approximation algorithm that showed how you can tackle heterogeneous computer scheduling (unrelated parallel machines) with linear programming, and still matches the best approximation factor to this date (some others have been proposed with the same approximation factor).  New results still draw from this paper to this date (see the cited in, there are a lot).

The notion of convexity can be applied to optimization problems and give us a better understanding of how well or just how we can approximate intractable problems.

Hope this helps :)!

This is a parameteriziation problem.

First consider the functions of 0 < t < 1

w1(t) = (8, 5, 12)*t

w2(t) = (10, 7, 17)*t

w3(t) = *5, 4, 7)* t

Plotting these points for 0 < t < 1 will give you the three vectors that you have in your diagram.

Next, for each 0 < t < 1, plot the lines w1(t) to w2(t); w2(t) to w3(t); w3(t) to w1(t).

This will give you triangles around the cone, so that you will see the surfaces as opaque.

Try plotting in 3D. If your software is smart enough, it should be smart enough to take care of hidden surfaces.

A large set of open problems is offered by the theory of quasiconvexity, that can be viewed as a continuous limit of the mathematical programming problem with infinitely many linear constraints. In other words, the minimizers - a gradient e=grad u or a y=curl v - satisfy differential constraints everywhere, curl y=0 and div y=0, respectively; these are the limits of difference constraints.