A Krasnosel'skii theorem for open finitely starlike sets in R 3

Let S be an open set in R ³ such that: (1) whenever three points x, y, z in S see each other via S, then conv{x, y, z} S, and (2) every seven points in S see via S a common point. Then S is finitely starlike. The proof uses the topological version of Helly's theorem.     

Some Krasnosel'skii numbers for finitely starlike sets in the plane

Let S be a subset of the plane. In case (int cl S) S = , then S is finitely starlike if and only if every 4 points of S see via S a common point. In case (int cl S) S has at most countably many components, each a singleton set, then S is finitely starlike if and only if every 5 points of S see via S a common point. Each of the numbers 4 and 5 is best possible. Examples show that these results fail without suitable restrictions on (int cl S) S. Moreover, a final example shows that if a general Krasnosel'skii number . exists to characterize finitely starlike sets in the plane, then > 9.     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

Sets with convex closure which are unions of two starshaped sets and families of segments which have a 2-partition

The following Krasnosel'skii-type theorem is proved: Let S be a nonempty set in R² whose closure cl S is convex and bounded. Assume that for every 9 point subset T of cl S there correspond points p₁ and p₂ (depending on T) such that each point of T is clearly visible via S from at least one of p₁ or p₂. Then S is a union of two starshaped sets. The number 9 is best possible.Moreover, a related result yields a piercing number for families of segments in R^d: Let £ be a collection of at least 6 one-dimensional convex sets in R^d such that for every line M in R^d, at most finitely many members of £ are collinear with M. Assume that every 6 members of £ may be partitioned into two sets £₁ and £₂ so that L L in £_i , i = 1,2. Then £ itself has such a 2-partition. The number 6 is best possible as well.Dedicated to Professor M. Barner on his 65^th birthday     

Points of local nonconvexity and sets which are almost starshaped

Let Sø be a bounded connected set in R ², and assume that every 3 or fewer lnc points of S are clearly visible from a common point of S. Then for some point p in S, the set A{s : s in S and [p, s] S} is nowhere dense in S. Furthermore, when S is open, then S in starshaped.     

A characterization theorem for certain unions of two starshaped sets in R 2

Let S be a compact set in R ². For S simply connected, S is a union of two starshaped sets if and only if for every F finite, F bdry S, there exist a set G bdry S arbitrarily close to F and points s, t depending on G such that each point of G is clearly visible via S from one of s, t. In the case where S has at most finitely many components, the necessity of the condition still holds while the sufficiency fails.     

A Chiral Lanthanide Tag for Stable and Rigid Attachment to Single Cysteine Residues in Proteins for NMR,EPR and Time-Resolved Luminescence Studies

A lanthanide-binding tag site-specifically attached to a protein presents a tool to probe the protein by multiple spectroscopic techniques, including nuclear magnetic resonance, electron paramagnetic resonance and time-resolved luminescence spectroscopy. Here a new stable chiral Ln^III tag, referred to as C12 , is presented for spontaneous and quantitative reaction with a cysteine residue to generate a stable thioether bond. The synthetic protocol of the tag is relatively straightforward, and the tag is stable for storage and shipping. It displays greatly enhanced reactivity towards selenocysteine, opening a route towards selective tagging of selenocysteine in proteins containing cysteine residues. Loaded with Tb^III or Tm^III ions, the C12 tag readily generates pseudocontact shifts (PCS) in protein NMR spectra. It produces a relatively rigid tether between lanthanide and protein, which is beneficial for interpretation of the PCSs by single magnetic susceptibility anisotropy tensors, and it is suitable for measuring distance distributions in double electron–electron resonance experiments. Upon reaction with cysteine or other thiol compounds, the Tb^III complex exhibits a 100-fold enhancement in luminescence quantum yield, affording a highly sensitive turn-on luminescence probe for time-resolved FRET assays and enzyme reaction monitoring.