Optimal shape of a blob

Abstract

This paper presents the solution to the following optimization problem: What is the shape of the two-dimensional region that minimizes the average L p distance between all pairs of points if the area of this region is held fixed? [The L p distance between two points x = (x 1 , x 2 ) andVariational techniques are used to show that the boundary curve of the optimal region satisfies a nonlinear integral equation. The special case p = 2 is elementary and for this case the integral equation reduces to a differential equation whose solution is a circle. Two nontrivial special cases, p = 1 and p = ∞, have already been examined in the literature. For these two cases the integral equation reduces to nonlinear secondorder differential equations, one of which contains a quadratic nonlinearity and the other a cubic nonlinearity.

City shapes that maximize the number of walking-only trips based on Manhattan distance

“…Bender et al (2004) proposed a similar problem focusing on the Manhattan distance, and they presented city shapes based on the variational method. Their method has been further extended so that it can be used to calculate optimal shapes based on a general L p metric (Bender et al 2007). These are treated as problems in continuous space, but some studies have discussed similar problems in a discrete framework (Avin et al 2015;Bender et al 2008;Demaine et al 2011;Fekete et al 2014).…”

Αλγόριθμοι Αναγνώρισης Και Επεξεργασίας Ακανόνιστων Περιοχών Ψηφιακής Εικόνας Με Τη Βοήθεια Κύκλου Και Έλλειψης

“…γηα ηελ εύξεζε πξώησλ αξηζκώλ, ν κεραληζκόο ησλ Αληηθπζήξσλ (87 π.Υ.) έλαο πνιπζύλζεηνο κεραληζκόο κε ηξηάληα δύν (32) νδνλησηά γξαλάδηα γηα αζηξνλνκηθέο εθηηκήζεηο, νη αιγόξηζκνη πνπ πεξηέρνληαη ζηα ζηνηρεία ηνπ Δπθιείδε (εύξεζε κέγηζηνπ θνηλνύ δηαηξέηε) θαη επηπιένλ αξθεηέο κέζνδνη γηα θάζε είδνπο ππνινγηζκνύο. Έθηνηε αξθεηνί επηζηήκνλεο κέρξη ζήκεξα, ζηεξίρηεθαλ ζηηο κειέηεο πξνγελέζηεξσλ εηώλ γηα ηε δηαηύπσζε θαη ηεθκεξίσζε λέσλ επηζηεκνληθώλ επηηεπγκάησλ.…”