The Determination of a Convex Set from Its Angle Function

Abstract

Abstract. In this paper we discuss the following question: how can we decide whether a convex set is determined by its angle function or not? We give sufficient conditions for convex polygons and for regular convex sets which guarantee that the set is distinguishable. We also investigate the question: which sets are typical (in the sense of Baire category), those which are distinguishable or those which are not? We prove that the family of distinguishable sets is of second Baire category.

Can you see the bubbles in a foam?

“…In Section 3 we first calculate the directional derivatives of the masking function, which generalizes a number of earlier results on visual angles given in [6][7][8] and [10][11][12][13]. At the end, the Laplacian of the masking function is given for d-weighted masking functions (Theorem 3.5).…”

“…We reformulate our question above as a problem to find such sets S of sources and sets O of objects that O T → M T ,ω S is injective. There are numerous results of this kind for special multicurves in the literature: [6][7][8], [10][11][12][13], and also [2,10,16] can be listed 6 here. With these terms the author's original Conjecture in [10] takes the following more general form.…”

“…In Section 4 the results 7 In Section 5 the masking function is supposed to be known only on an open ring around the trace of a multicurve. First, it is shown that multisegments can be distinguished from each other by this restricted knowledge (Theorem 5.1).…”

Finding Needles in a Haystack

“…However it is proved in [6,8], that two polygons are always distinguishable from each other, so the question Are convex polygons distinguishable among convex domains? (1) emerged naturally in [4,Quest. 3.2], where it is proved that all triangles [4,Cor.…”

Visual distinguishability of polygons

“…The central problem in this subject is to show a set S of sources and a set O of objects that O B → ν B S is injective. There are a number of such results in the literature; see [3], [4], [5], [6], [7] etc.…”